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n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
this : DecidableEq L := Classical.decEq L
⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
| obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n) | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
| Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
this : DecidableEq L := Classical.decEq L
ζ : L
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin K (rootSet (cyclotomic (↑n) K) L) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
| exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
| Mathlib.NumberTheory.Cyclotomic.Basic.497_0.xReI1DeVvechFQU | /-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Field (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
| delta CyclotomicField | instance : Field (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.528_0.xReI1DeVvechFQU | instance : Field (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Field (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; | infer_instance | instance : Field (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.528_0.xReI1DeVvechFQU | instance : Field (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Algebra K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
| delta CyclotomicField | instance algebra : Algebra K (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.532_0.xReI1DeVvechFQU | instance algebra : Algebra K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Algebra K (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; | infer_instance | instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.532_0.xReI1DeVvechFQU | instance algebra : Algebra K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Inhabited (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
| delta CyclotomicField | instance : Inhabited (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.536_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
⊢ Inhabited (SplittingField (cyclotomic (↑n) K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; | infer_instance | instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; | Mathlib.NumberTheory.Cyclotomic.Basic.536_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
| haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
| letI := Classical.decEq (CyclotomicField n K) | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
| obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne' | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : eval₂ (algebraMap K (CyclotomicField n K)) ζ (cyclotomic (↑n) K) = 0
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
| rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : IsPrimitiveRoot ζ ↑n
⊢ IsCyclotomicExtension {n} K (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
| refine ⟨?_, ?_⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : IsPrimitiveRoot ζ ↑n
⊢ ∀ {n_1 : ℕ+}, n_1 ∈ {n} → ∃ r, IsPrimitiveRoot r ↑n_1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· | simp only [mem_singleton_iff, forall_eq] | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : IsPrimitiveRoot ζ ↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
| exact ⟨ζ, hζ⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : IsPrimitiveRoot ζ ↑n
⊢ ∀ (x : CyclotomicField n K), x ∈ adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· | rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm] | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· | Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
ζ : CyclotomicField n K
hζ : IsPrimitiveRoot ζ ↑n
⊢ adjoin K (rootSet (cyclotomic (↑n) K) (SplittingField (cyclotomic (↑n) K))) = adjoin K {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
| exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
| Mathlib.NumberTheory.Cyclotomic.Basic.542_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ NoZeroSMulDivisors A (CyclotomicField n K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
| refine' NoZeroSMulDivisors.of_algebraMap_injective _ | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Function.Injective ⇑(algebraMap A (CyclotomicField n K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
| rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)] | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Function.Injective ⇑(RingHom.comp (algebraMap K (CyclotomicField n K)) (algebraMap A K)) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
| exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _) | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
| Mathlib.NumberTheory.Cyclotomic.Basic.588_0.xReI1DeVvechFQU | instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ CommRing (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
| delta CyclotomicRing | instance : CommRing (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.608_0.xReI1DeVvechFQU | instance : CommRing (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ CommRing ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; | infer_instance | instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.608_0.xReI1DeVvechFQU | instance : CommRing (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ IsDomain (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
| delta CyclotomicRing | instance : IsDomain (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.612_0.xReI1DeVvechFQU | instance : IsDomain (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ IsDomain ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; | infer_instance | instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.612_0.xReI1DeVvechFQU | instance : IsDomain (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Inhabited (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
| delta CyclotomicRing | instance : Inhabited (CyclotomicRing n A K) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.616_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Inhabited ↥(adjoin A {b | b ^ ↑n = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; | infer_instance | instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; | Mathlib.NumberTheory.Cyclotomic.Basic.616_0.xReI1DeVvechFQU | instance : Inhabited (CyclotomicRing n A K) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
a : ℕ+
han : a ∈ {n}
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
| rw [mem_singleton_iff] at han | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
a : ℕ+
han : a = n
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
| subst a | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
| haveI := NeZero.of_noZeroSMulDivisors A K n | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
| haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
| obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n) | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
hμ : IsPrimitiveRoot μ ↑n
⊢ ∃ r, IsPrimitiveRoot r ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
| refine' ⟨⟨μ, subset_adjoin _⟩, _⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
hμ : IsPrimitiveRoot μ ↑n
⊢ μ ∈ {b | b ^ ↑n = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· | apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
hμ : IsPrimitiveRoot μ ↑n
⊢ ∃ i ∈ Nat.divisors ↑n, IsRoot (cyclotomic i (CyclotomicField n K)) μ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
| refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
hμ : IsPrimitiveRoot μ ↑n
⊢ IsRoot (cyclotomic (↑n) (CyclotomicField n K)) μ | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
| rwa [← isRoot_cyclotomic_iff] at hμ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro.refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
this✝ : NeZero ↑↑n
this : NeZero ↑↑n
μ : CyclotomicField n K
hμ : IsPrimitiveRoot μ ↑n
⊢ IsPrimitiveRoot { val := μ, property := (_ : μ ∈ ↑(adjoin A {b | b ^ ↑n = 1})) } ↑n | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· | rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
⊢ x ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
| refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ { val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) } ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· | refine' subset_adjoin _ | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ { val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) } ∈ {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
| simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ { val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) } ^ ↑n = 1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
| rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk] | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
| Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
a : A
⊢ (algebraMap A ↥(adjoin A {b | b ^ ↑n = 1})) a ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· | exact Subalgebra.algebraMap_mem _ a | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_3
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y z : ↥(adjoin A {b | b ^ ↑n = 1})
hy : y ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
hz : z ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
⊢ y + z ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· | exact Subalgebra.add_mem _ hy hz | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine'_4
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑↑n
x : CyclotomicRing n A K
y z : ↥(adjoin A {b | b ^ ↑n = 1})
hy : y ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
hz : z ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1}
⊢ y * z ∈ adjoin A {b | ∃ n_1 ∈ {n}, b ^ ↑n_1 = 1} | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· | exact Subalgebra.mul_mem _ hy hz | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· | Mathlib.NumberTheory.Cyclotomic.Basic.649_0.xReI1DeVvechFQU | instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ IsUnit ((algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑{ val := x, property := hx }) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
| rw [isUnit_iff_ne_zero] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑{ val := x, property := hx } ≠ 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
| apply map_ne_zero_of_mem_nonZeroDivisors | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case hg
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ Function.Injective ⇑(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
case h
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ ↑{ val := x, property := hx } ∈ nonZeroDivisors (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
| apply adjoin_algebra_injective | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case h
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ ↑{ val := x, property := hx } ∈ nonZeroDivisors (CyclotomicRing n A K) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
| exact hx | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
⊢ ∃ x_1,
x * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
| letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K) | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
⊢ ∃ x_1,
x * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
| refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_1
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ ∃ x,
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· | exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ y *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
↑({ val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) }, 1).2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
({ val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) }, 1).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by | simp | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y : CyclotomicField n K
hy : y ∈ {b | b ^ ↑n = 1}
⊢ y =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
{ val := y, property := (_ : y ∈ ↑(adjoin A {b | b ^ ↑n = 1})) } | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; | rfl | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
⊢ ∃ x,
(algebraMap K (CyclotomicField n K)) k * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· | have : IsLocalization (nonZeroDivisors A) K := inferInstance | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
this : IsLocalization (nonZeroDivisors A) K
⊢ ∃ x,
(algebraMap K (CyclotomicField n K)) k * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
| replace := this.surj | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
this : ∀ (z : K), ∃ x, z * (algebraMap A K) ↑x.2 = (algebraMap A K) x.1
⊢ ∃ x,
(algebraMap K (CyclotomicField n K)) k * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
| obtain ⟨⟨z, w⟩, hw⟩ := this k | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
this : ∀ (z : K), ∃ x, z * (algebraMap A K) ↑x.2 = (algebraMap A K) x.1
z : A
w : ↥(nonZeroDivisors A)
hw : k * (algebraMap A K) ↑(z, w).2 = (algebraMap A K) (z, w).1
⊢ ∃ x,
(algebraMap K (CyclotomicField n K)) k * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
| refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝ : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
this : ∀ (z : K), ∃ x, z * (algebraMap A K) ↑x.2 = (algebraMap A K) x.1
z : A
w : ↥(nonZeroDivisors A)
hw : k * (algebraMap A K) ↑(z, w).2 = (algebraMap A K) (z, w).1
⊢ (algebraMap K (CyclotomicField n K)) k *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
↑((algebraMap A (CyclotomicRing n A K)) z,
{ val := (algebraMap A (CyclotomicRing n A K)) ↑w,
property :=
(_ : (algebraMap A (CyclotomicRing n A K)) ↑w ∈ nonZeroDivisors (CyclotomicRing n A K)) }).2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
((algebraMap A (CyclotomicRing n A K)) z,
{ val := (algebraMap A (CyclotomicRing n A K)) ↑w,
property := (_ : (algebraMap A (CyclotomicRing n A K)) ↑w ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
| letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl) | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_2.intro.mk
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this✝¹ : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
k : K
this✝ : ∀ (z : K), ∃ x, z * (algebraMap A K) ↑x.2 = (algebraMap A K) x.1
z : A
w : ↥(nonZeroDivisors A)
hw : k * (algebraMap A K) ↑(z, w).2 = (algebraMap A K) (z, w).1
this : IsScalarTower A K (CyclotomicField n K) := IsScalarTower.of_algebraMap_eq (congr_fun rfl)
⊢ (algebraMap K (CyclotomicField n K)) k *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
↑((algebraMap A (CyclotomicRing n A K)) z,
{ val := (algebraMap A (CyclotomicRing n A K)) ↑w,
property :=
(_ : (algebraMap A (CyclotomicRing n A K)) ↑w ∈ nonZeroDivisors (CyclotomicRing n A K)) }).2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
((algebraMap A (CyclotomicRing n A K)) z,
{ val := (algebraMap A (CyclotomicRing n A K)) ↑w,
property := (_ : (algebraMap A (CyclotomicRing n A K)) ↑w ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
| rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
⊢ ∀ (x y : CyclotomicField n K),
(∃ x_1,
x * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1) →
(∃ x,
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1) →
∃ x_1,
(x + y) * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· | rintro y z ⟨a, ha⟩ ⟨b, hb⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ ∃ x,
(y + z) * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
| refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ (y + z) *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
↑(a.1 * ↑b.2 + b.1 * ↑a.2,
{ val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
(a.1 * ↑b.2 + b.1 * ↑a.2,
{ val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
| rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_3.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1 *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 +
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1 *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
(a.1 * ↑b.2 + b.1 * ↑a.2,
{ val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
| simp only [map_add, map_mul] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
⊢ ∀ (x y : CyclotomicField n K),
(∃ x_1,
x * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1) →
(∃ x,
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1) →
∃ x_1,
x * y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x_1.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· | rintro y z ⟨a, ha⟩ ⟨b, hb⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ ∃ x,
y * z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
| refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩ | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ y * z *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
↑(a.1 * b.1, { val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
(a.1 * b.1, { val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
| rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
case refine_4.intro.intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x : CyclotomicField n K
this : NeZero ↑↑n := NeZero.nat_of_injective (IsFractionRing.injective A K)
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1 *
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1 =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))
(a.1 * b.1, { val := ↑a.2 * ↑b.2, property := (_ : ↑a.2 * ↑b.2 ∈ nonZeroDivisors (CyclotomicRing n A K)) }).1 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
| simp only [map_mul] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
| Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑↑n
x y : CyclotomicRing n A K
h :
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x =
(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) y
⊢ ↑1 * x = ↑1 * y | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by | rw [adjoin_algebra_injective n A K h] | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by | Mathlib.NumberTheory.Cyclotomic.Basic.673_0.xReI1DeVvechFQU | instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
μ : CyclotomicField n K
h : IsPrimitiveRoot μ ↑n
⊢ CyclotomicRing n A K = ↥(adjoin A {μ}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
| rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h] | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
| Mathlib.NumberTheory.Cyclotomic.Basic.712_0.xReI1DeVvechFQU | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
μ : CyclotomicField n K
h : IsPrimitiveRoot μ ↑n
⊢ CyclotomicRing n A K = ↥(adjoin A {b | ∃ a ∈ {n}, b ^ ↑a = 1}) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
| simp [CyclotomicRing] | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
| Mathlib.NumberTheory.Cyclotomic.Basic.712_0.xReI1DeVvechFQU | theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
⊢ IsCyclotomicExtension S K K | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
simp [CyclotomicRing]
#align cyclotomic_ring.eq_adjoin_primitive_root CyclotomicRing.eq_adjoin_primitive_root
end CyclotomicRing
end CyclotomicRing
end IsDomain
section IsAlgClosed
variable [IsAlgClosed K]
/-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
| refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩ | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
simp [CyclotomicRing]
#align cyclotomic_ring.eq_adjoin_primitive_root CyclotomicRing.eq_adjoin_primitive_root
end CyclotomicRing
end CyclotomicRing
end IsDomain
section IsAlgClosed
variable [IsAlgClosed K]
/-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
| obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne' | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
⊢ ∃ r, IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
simp [CyclotomicRing]
#align cyclotomic_ring.eq_adjoin_primitive_root CyclotomicRing.eq_adjoin_primitive_root
end CyclotomicRing
end CyclotomicRing
end IsDomain
section IsAlgClosed
variable [IsAlgClosed K]
/-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
| refine' ⟨r, _⟩ | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
⊢ IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
simp [CyclotomicRing]
#align cyclotomic_ring.eq_adjoin_primitive_root CyclotomicRing.eq_adjoin_primitive_root
end CyclotomicRing
end CyclotomicRing
end IsDomain
section IsAlgClosed
variable [IsAlgClosed K]
/-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
refine' ⟨r, _⟩
| haveI := h a ha | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
refine' ⟨r, _⟩
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsAlgClosed K
h : ∀ a ∈ S, NeZero ↑↑a
a : ℕ+
ha : a ∈ S
r : K
hr : (aeval r) (cyclotomic (↑a) K) = 0
this : NeZero ↑↑a
⊢ IsPrimitiveRoot r ↑a | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
/-!
# Cyclotomic extensions
Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ+`, we define a class
`IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th
primitive roots of unity, for all `n ∈ S`.
## Main definitions
* `IsCyclotomicExtension S A B` : means that `B` is obtained from `A` by adding `n`-th primitive
roots of unity, for all `n ∈ S`.
* `CyclotomicField`: given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has the instance
`IsCyclotomicExtension {n} K (CyclotomicField n K)`.
* `CyclotomicRing` : if `A` is a domain with fraction field `K` and `n : ℕ+`, we define
`CyclotomicRing n A K` as the `A`-subalgebra of `CyclotomicField n K` generated by the roots of
`X ^ n - 1`. If `n` is nonzero in `A`, it has the instance
`IsCyclotomicExtension {n} A (CyclotomicRing n A K)`.
## Main results
* `IsCyclotomicExtension.trans` : if `IsCyclotomicExtension S A B` and
`IsCyclotomicExtension T B C`, then `IsCyclotomicExtension (S ∪ T) A C` if
`Function.Injective (algebraMap B C)`.
* `IsCyclotomicExtension.union_right` : given `IsCyclotomicExtension (S ∪ T) A B`, then
`IsCyclotomicExtension T (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }) B`.
* `IsCyclotomicExtension.union_left` : given `IsCyclotomicExtension T A B` and `S ⊆ T`, then
`IsCyclotomicExtension S A (adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 })`.
* `IsCyclotomicExtension.finite` : if `S` is finite and `IsCyclotomicExtension S A B`, then
`B` is a finite `A`-algebra.
* `IsCyclotomicExtension.numberField` : a finite cyclotomic extension of a number field is a
number field.
* `IsCyclotomicExtension.splitting_field_X_pow_sub_one` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `X ^ n - 1`.
* `IsCyclotomicExtension.splitting_field_cyclotomic` : if `IsCyclotomicExtension {n} K L`,
then `L` is the splitting field of `cyclotomic n K`.
## Implementation details
Our definition of `IsCyclotomicExtension` is very general, to allow rings of any characteristic
and infinite extensions, but it will mainly be used in the case `S = {n}` and for integral domains.
All results are in the `IsCyclotomicExtension` namespace.
Note that some results, for example `IsCyclotomicExtension.trans`,
`IsCyclotomicExtension.finite`, `IsCyclotomicExtension.numberField`,
`IsCyclotomicExtension.finiteDimensional`, `IsCyclotomicExtension.isGalois` and
`CyclotomicField.algebraBase` are lemmas, but they can be made local instances. Some of them are
included in the `Cyclotomic` locale.
-/
open Polynomial Algebra FiniteDimensional Set
open scoped BigOperators
universe u v w z
variable (n : ℕ+) (S T : Set ℕ+) (A : Type u) (B : Type v) (K : Type w) (L : Type z)
variable [CommRing A] [CommRing B] [Algebra A B]
variable [Field K] [Field L] [Algebra K L]
noncomputable section
/-- Given an `A`-algebra `B` and `S : Set ℕ+`, we define `IsCyclotomicExtension S A B` requiring
that there is an `n`-th primitive root of unity in `B` for all `n ∈ S` and that `B` is generated
over `A` by the roots of `X ^ n - 1`. -/
@[mk_iff]
class IsCyclotomicExtension : Prop where
/-- For all `n ∈ S`, there exists a primitive `n`-th root of unity in `B`. -/
exists_prim_root {n : ℕ+} (ha : n ∈ S) : ∃ r : B, IsPrimitiveRoot r n
/-- The `n`-th roots of unity, for `n ∈ S`, generate `B` as an `A`-algebra. -/
adjoin_roots : ∀ x : B, x ∈ adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}
#align is_cyclotomic_extension IsCyclotomicExtension
namespace IsCyclotomicExtension
section Basic
/-- A reformulation of `IsCyclotomicExtension` that uses `⊤`. -/
theorem iff_adjoin_eq_top :
IsCyclotomicExtension S A B ↔
(∀ n : ℕ+, n ∈ S → ∃ r : B, IsPrimitiveRoot r n) ∧
adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} = ⊤ :=
⟨fun h => ⟨fun _ => h.exists_prim_root, Algebra.eq_top_iff.2 h.adjoin_roots⟩, fun h =>
⟨h.1 _, Algebra.eq_top_iff.1 h.2⟩⟩
#align is_cyclotomic_extension.iff_adjoin_eq_top IsCyclotomicExtension.iff_adjoin_eq_top
/-- A reformulation of `IsCyclotomicExtension` in the case `S` is a singleton. -/
theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} :=
by simp [IsCyclotomicExtension_iff]
#align is_cyclotomic_extension.iff_singleton IsCyclotomicExtension.iff_singleton
/-- If `IsCyclotomicExtension ∅ A B`, then the image of `A` in `B` equals `B`. -/
theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by
simpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h
#align is_cyclotomic_extension.empty IsCyclotomicExtension.empty
/-- If `IsCyclotomicExtension {1} A B`, then the image of `A` in `B` equals `B`. -/
theorem singleton_one [h : IsCyclotomicExtension {1} A B] : (⊥ : Subalgebra A B) = ⊤ :=
Algebra.eq_top_iff.2 fun x => by
simpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x
#align is_cyclotomic_extension.singleton_one IsCyclotomicExtension.singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension ∅ A B`. -/
theorem singleton_zero_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension ∅ A B := by
-- Porting note: Lean3 is able to infer `A`.
refine' (iff_adjoin_eq_top _ A _).2
⟨fun s hs => by simp at hs, _root_.eq_top_iff.2 fun x hx => _⟩
rw [← h] at hx
simpa using hx
#align is_cyclotomic_extension.singleton_zero_of_bot_eq_top IsCyclotomicExtension.singleton_zero_of_bot_eq_top
variable (A B)
/-- Transitivity of cyclotomic extensions. -/
theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by
refine' ⟨fun hn => _, fun x => _⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine' ⟨algebraMap B C b, _⟩
exact hb.map_of_injective h
· exact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine' adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)
(fun c ⟨n, hn⟩ => subset_adjoin ⟨n, Or.inr hn.1, hn.2⟩) (fun b => _)
(fun x y hx hy => Subalgebra.add_mem _ hx hy) fun x y hx hy => Subalgebra.mul_mem _ hx hy
· let f := IsScalarTower.toAlgHom A B C
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}).map f :=
⟨b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
rw [IsScalarTower.toAlgHom_apply, ← adjoin_image] at hb
refine' adjoin_mono (fun y hy => _) hb
obtain ⟨b₁, ⟨⟨n, hn⟩, h₁⟩⟩ := hy
exact ⟨n, ⟨mem_union_left T hn.1, by rw [← h₁, ← AlgHom.map_pow, hn.2, AlgHom.map_one]⟩⟩
#align is_cyclotomic_extension.trans IsCyclotomicExtension.trans
@[nontriviality]
theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porting note: `R := A` was not needed.
· exact ⟨fun h => h.elim, fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
· rw [iff_singleton]
exact ⟨⟨0, IsPrimitiveRoot.of_subsingleton 0⟩,
fun x => by convert (mem_top (R := A) : x ∈ ⊤)⟩
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine' le_antisymm _ _
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintro x (⟨n, hn⟩ | ⟨n, hn⟩)
· exact ⟨n, Or.inl hn.1, hn.2⟩
· exact ⟨n, Or.inr hn.1, hn.2⟩
refine' ⟨fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _⟩
replace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b
rwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h
#align is_cyclotomic_extension.union_right IsCyclotomicExtension.union_right
/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `T` and `S ⊆ T`,
then `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` is a cyclotomic extension of `B`
given by roots of unity of order in `S`. -/
theorem union_left [h : IsCyclotomicExtension T A B] (hS : S ⊆ T) :
IsCyclotomicExtension S A (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) := by
refine' ⟨@fun n hn => _, fun b => _⟩
· obtain ⟨b, hb⟩ := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)
refine' ⟨⟨b, subset_adjoin ⟨n, hn, hb.pow_eq_one⟩⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
· convert mem_top (R := A) (x := b)
rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq]
norm_cast
#align is_cyclotomic_extension.union_left IsCyclotomicExtension.union_left
variable {n S}
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` implies
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem of_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) [H : IsCyclotomicExtension S A B] :
IsCyclotomicExtension (S ∪ {n}) A B := by
refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩
· rw [mem_union, mem_singleton_iff] at hs
obtain hs | rfl := hs
· exact H.exists_prim_root hs
· obtain ⟨m, hm⟩ := hS
obtain ⟨x, rfl⟩ := h m hm
obtain ⟨ζ, hζ⟩ := H.exists_prim_root hm
refine' ⟨ζ ^ (x : ℕ), _⟩
convert hζ.pow_of_dvd x.ne_zero (dvd_mul_left (x : ℕ) s)
simp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]
· refine' _root_.eq_top_iff.2 _
rw [← ((iff_adjoin_eq_top S A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, hm⟩ := hx
exact ⟨m, ⟨Or.inr hm.1, hm.2⟩⟩
#align is_cyclotomic_extension.of_union_of_dvd IsCyclotomicExtension.of_union_of_dvd
/-- If `∀ s ∈ S, n ∣ s` and `S` is not empty, then `IsCyclotomicExtension S A B` if and only if
`IsCyclotomicExtension (S ∪ {n}) A B`. -/
theorem iff_union_of_dvd (h : ∀ s ∈ S, n ∣ s) (hS : S.Nonempty) :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {n}) A B := by
refine'
⟨fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => _, _⟩⟩
· exact H.exists_prim_root (subset_union_left _ _ hs)
· rw [_root_.eq_top_iff, ← ((iff_adjoin_eq_top _ A B).1 H).2]
refine' adjoin_mono fun x hx => _
simp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx ⊢
obtain ⟨m, rfl | hm, hxpow⟩ := hx
· obtain ⟨y, hy⟩ := hS
refine' ⟨y, ⟨hy, _⟩⟩
obtain ⟨z, rfl⟩ := h y hy
simp only [PNat.mul_coe, pow_mul, hxpow, one_pow]
· exact ⟨m, ⟨hm, hxpow⟩⟩
#align is_cyclotomic_extension.iff_union_of_dvd IsCyclotomicExtension.iff_union_of_dvd
variable (n S)
/-- `IsCyclotomicExtension S A B` is equivalent to `IsCyclotomicExtension (S ∪ {1}) A B`. -/
theorem iff_union_singleton_one :
IsCyclotomicExtension S A B ↔ IsCyclotomicExtension (S ∪ {1}) A B := by
obtain hS | rfl := S.eq_empty_or_nonempty.symm
· exact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS
rw [empty_union]
refine' ⟨fun H => _, fun H => _⟩
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => ⟨1, by simp [mem_singleton_iff.1 hs]⟩, _⟩
simp [adjoin_singleton_one, empty]
· refine' (iff_adjoin_eq_top _ A _).2 ⟨fun s hs => (not_mem_empty s hs).elim, _⟩
simp [@singleton_one A B _ _ _ H]
#align is_cyclotomic_extension.iff_union_singleton_one IsCyclotomicExtension.iff_union_singleton_one
variable {A B}
/-- If `(⊥ : SubAlgebra A B) = ⊤`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_bot_eq_top (h : (⊥ : Subalgebra A B) = ⊤) :
IsCyclotomicExtension {1} A B := by
convert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)
simp
#align is_cyclotomic_extension.singleton_one_of_bot_eq_top IsCyclotomicExtension.singleton_one_of_bot_eq_top
/-- If `Function.Surjective (algebraMap A B)`, then `IsCyclotomicExtension {1} A B`. -/
theorem singleton_one_of_algebraMap_bijective (h : Function.Surjective (algebraMap A B)) :
IsCyclotomicExtension {1} A B :=
singleton_one_of_bot_eq_top (surjective_algebraMap_iff.1 h).symm
#align is_cyclotomic_extension.singleton_one_of_algebra_map_bijective IsCyclotomicExtension.singleton_one_of_algebraMap_bijective
variable (A B)
/-- Given `(f : B ≃ₐ[A] C)`, if `IsCyclotomicExtension S A B` then
`IsCyclotomicExtension S A C`. -/
protected
theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
#align is_cyclotomic_extension.equiv IsCyclotomicExtension.equiv
protected
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B) := by
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
exact hr.neZero'
#align is_cyclotomic_extension.ne_zero IsCyclotomicExtension.neZero
protected
theorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : A) := by
haveI := IsCyclotomicExtension.neZero n A B
exact NeZero.nat_of_neZero (algebraMap A B)
#align is_cyclotomic_extension.ne_zero' IsCyclotomicExtension.neZero'
end Basic
section Fintype
theorem finite_of_singleton [IsDomain B] [h : IsCyclotomicExtension {n} A B] :
Module.Finite A B := by
classical
rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2]
refine' fg_adjoin_of_finite _ fun b hb => _
· simp only [mem_singleton_iff, exists_eq_left]
have : {b : B | b ^ (n : ℕ) = 1} = (nthRoots n (1 : B)).toFinset :=
Set.ext fun x => ⟨fun h => by simpa using h, fun h => by simpa using h⟩
rw [this]
exact (nthRoots (↑n) 1).toFinset.finite_toSet
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb
refine' ⟨X ^ (n : ℕ) - 1, ⟨monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]⟩⟩
#align is_cyclotomic_extension.finite_of_singleton IsCyclotomicExtension.finite_of_singleton
/-- If `S` is finite and `IsCyclotomicExtension S A B`, then `B` is a finite `A`-algebra. -/
protected
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B := by
cases' nonempty_fintype S with h
revert h₂ A B
refine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _
· intro _ _ _ _ _
refine' Module.finite_def.2 ⟨({1} : Finset B), _⟩
simp [← top_toSubmodule, ← empty, toSubmodule_bot, Submodule.one_eq_span]
· intro _ _ _ _ h
haveI : IsCyclotomicExtension S A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) :=
union_left _ (insert n S) _ _ (subset_insert n S)
haveI := H A (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1})
have : Module.Finite (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) B := by
rw [← union_singleton] at h
letI := @union_right S {n} A B _ _ _ h
exact finite_of_singleton n _ _
exact Module.Finite.trans (adjoin A {b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1}) _
#align is_cyclotomic_extension.finite IsCyclotomicExtension.finite
/-- A cyclotomic finite extension of a number field is a number field. -/
theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L :=
{ to_charZero := charZero_of_injective_algebraMap (algebraMap K L).injective
to_finiteDimensional := by
haveI := charZero_of_injective_algebraMap (algebraMap K L).injective
haveI := IsCyclotomicExtension.finite S K L
exact Module.Finite.trans K _ }
#align is_cyclotomic_extension.number_field IsCyclotomicExtension.numberField
/-- A finite cyclotomic extension of an integral noetherian domain is integral -/
theorem integral [IsDomain B] [IsNoetherianRing A] [Finite S] [IsCyclotomicExtension S A B] :
Algebra.IsIntegral A B :=
letI := IsCyclotomicExtension.finite S A B; isIntegral_of_noetherian inferInstance
#align is_cyclotomic_extension.integral IsCyclotomicExtension.integral
/-- If `S` is finite and `IsCyclotomicExtension S K A`, then `finiteDimensional K A`. -/
theorem finiteDimensional (C : Type z) [Finite S] [CommRing C] [Algebra K C] [IsDomain C]
[IsCyclotomicExtension S K C] : FiniteDimensional K C :=
IsCyclotomicExtension.finite S K C
#align is_cyclotomic_extension.finite_dimensional IsCyclotomicExtension.finiteDimensional
end Fintype
section
variable {A B}
theorem adjoin_roots_cyclotomic_eq_adjoin_nth_roots [IsDomain B] {ζ : B} {n : ℕ+}
(hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)
· rw [mem_rootSet'] at hx
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rw [isRoot_of_unity_iff n.pos]
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [IsRoot.def, ← map_cyclotomic n (algebraMap A B), eval_map, ← aeval_def]
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)
rw [mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, map_cyclotomic, ← IsRoot]
refine' ⟨cyclotomic_ne_zero n B, hζ.isRoot_cyclotomic n.pos⟩
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots
theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by
refine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)
· suffices hx : x ^ n.1 = 1
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine' (isRoot_of_unity_iff n.pos B).2 _
refine' ⟨n, Nat.mem_divisors_self n n.ne_zero, _⟩
rw [mem_rootSet', aeval_def, ← eval_map, map_cyclotomic, ← IsRoot] at hx
exact hx.2
· simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx
simpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, ← eval_map, IsRoot] using
And.intro (cyclotomic_ne_zero n B) (hζ.isRoot_cyclotomic n.pos)
#align is_cyclotomic_extension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic
theorem adjoin_primitive_root_eq_top {n : ℕ+} [IsDomain B] [h : IsCyclotomicExtension {n} A B]
{ζ : B} (hζ : IsPrimitiveRoot ζ n) : adjoin A ({ζ} : Set B) = ⊤ := by
classical
rw [← adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic hζ]
rw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ]
exact ((iff_adjoin_eq_top {n} A B).mp h).2
#align is_cyclotomic_extension.adjoin_primitive_root_eq_top IsCyclotomicExtension.adjoin_primitive_root_eq_top
variable (A)
theorem _root_.IsPrimitiveRoot.adjoin_isCyclotomicExtension {ζ : B} {n : ℕ+}
(h : IsPrimitiveRoot ζ n) : IsCyclotomicExtension {n} A (adjoin A ({ζ} : Set B)) :=
{ exists_prim_root := fun hi => by
rw [Set.mem_singleton_iff] at hi
refine' ⟨⟨ζ, subset_adjoin <| Set.mem_singleton ζ⟩, _⟩
rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]
adjoin_roots := fun x => by
refine
adjoin_induction'
(x := x) (fun b hb => ?_) (fun a => ?_) (fun b₁ b₂ hb₁ hb₂ => ?_)
(fun b₁ b₂ hb₁ hb₂ => ?_)
· rw [Set.mem_singleton_iff] at hb
refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]
rw [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
exact ((IsPrimitiveRoot.iff_def ζ n).1 h).1
· exact Subalgebra.algebraMap_mem _ _
· exact Subalgebra.add_mem _ hb₁ hb₂
· exact Subalgebra.mul_mem _ hb₁ hb₂ }
#align is_primitive_root.adjoin_is_cyclotomic_extension IsPrimitiveRoot.adjoin_isCyclotomicExtension
end
section Field
variable {n S}
/-- A cyclotomic extension splits `X ^ n - 1` if `n ∈ S`.-/
theorem splits_X_pow_sub_one [H : IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (X ^ (n : ℕ) - 1) := by
rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X]
obtain ⟨z, hz⟩ := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS
exact X_pow_sub_one_splits hz
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splits_X_pow_sub_one IsCyclotomicExtension.splits_X_pow_sub_one
/-- A cyclotomic extension splits `cyclotomic n K` if `n ∈ S` and `ne_zero (n : K)`.-/
theorem splits_cyclotomic [IsCyclotomicExtension S K L] (hS : n ∈ S) :
Splits (algebraMap K L) (cyclotomic n K) := by
refine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _
use ∏ i : ℕ in (n : ℕ).properDivisors, Polynomial.cyclotomic i K
rw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]
#align is_cyclotomic_extension.splits_cyclotomic IsCyclotomicExtension.splits_cyclotomic
variable (n S)
section Singleton
variable [IsCyclotomicExtension {n} K L]
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `X ^ n - 1`. -/
theorem isSplittingField_X_pow_sub_one : IsSplittingField K L (X ^ (n : ℕ) - 1) :=
{ splits' := splits_X_pow_sub_one K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
congr
refine' Set.ext fun x => _
simp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left,
mem_setOf_eq, Polynomial.map_X, Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]
simp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X,
and_iff_right_iff_imp, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]
exact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L) }
set_option linter.uppercaseLean3 false in
#align is_cyclotomic_extension.splitting_field_X_pow_sub_one IsCyclotomicExtension.isSplittingField_X_pow_sub_one
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
(@IsSplittingField.algEquiv K L _ _ _ (X ^ (n : ℕ) - 1) h₁).trans
(@IsSplittingField.algEquiv K L' _ _ _ (X ^ (n : ℕ) - 1) h₂).symm
#align is_cyclotomic_extension.alg_equiv IsCyclotomicExtension.algEquiv
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.isSplittingField_X_pow_sub_one
theorem isGalois : IsGalois K L :=
letI := isSplittingField_X_pow_sub_one n K L
IsGalois.of_separable_splitting_field (X_pow_sub_one_separable_iff.2
(IsCyclotomicExtension.neZero' n K L).1)
#align is_cyclotomic_extension.is_galois IsCyclotomicExtension.isGalois
/-- If `IsCyclotomicExtension {n} K L`, then `L` is the splitting field of `cyclotomic n K`. -/
theorem splitting_field_cyclotomic : IsSplittingField K L (cyclotomic n K) :=
{ splits' := splits_cyclotomic K L (mem_singleton n)
adjoin_rootSet' := by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
-- todo: make `exists_prim_root` take an explicit `L`
obtain ⟨ζ : L, hζ⟩ := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ }
#align is_cyclotomic_extension.splitting_field_cyclotomic IsCyclotomicExtension.splitting_field_cyclotomic
scoped[Cyclotomic] attribute [instance] IsCyclotomicExtension.splitting_field_cyclotomic
end Singleton
end Field
end IsCyclotomicExtension
section CyclotomicField
/-- Given `n : ℕ+` and a field `K`, we define `CyclotomicField n K` as the
splitting field of `cyclotomic n K`. If `n` is nonzero in `K`, it has
the instance `IsCyclotomicExtension {n} K (CyclotomicField n K)`. -/
def CyclotomicField : Type w :=
(cyclotomic n K).SplittingField
#align cyclotomic_field CyclotomicField
namespace CyclotomicField
--Porting note: could not be derived
instance : Field (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance algebra : Algebra K (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicField n K) := by
delta CyclotomicField; infer_instance
instance [CharZero K] : CharZero (CyclotomicField n K) :=
charZero_of_injective_algebraMap (algebraMap K _).injective
instance isCyclotomicExtension [NeZero ((n : ℕ) : K)] :
IsCyclotomicExtension {n} K (CyclotomicField n K) := by
haveI : NeZero ((n : ℕ) : CyclotomicField n K) :=
NeZero.nat_of_injective (algebraMap K _).injective
letI := Classical.decEq (CyclotomicField n K)
obtain ⟨ζ, hζ⟩ :=
exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)
(degree_cyclotomic_pos n K n.pos).ne'
rw [← eval_map, ← IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at hζ
-- Porting note: the first `?_` was `forall_eq.2 ⟨ζ, hζ⟩` that now fails.
refine ⟨?_, ?_⟩
· simp only [mem_singleton_iff, forall_eq]
exact ⟨ζ, hζ⟩
· rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]
exact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ
#align cyclotomic_field.is_cyclotomic_extension CyclotomicField.isCyclotomicExtension
end CyclotomicField
end CyclotomicField
section IsDomain
variable [Algebra A K] [IsFractionRing A K]
section CyclotomicRing
/-- If `K` is the fraction field of `A`, the `A`-algebra structure on `CyclotomicField n K`.
-/
@[nolint unusedArguments]
instance CyclotomicField.algebraBase : Algebra A (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra_base CyclotomicField.algebraBase
/-- Ensure there are no diamonds when `A = ℤ`. -/
example : algebraInt (CyclotomicField n ℚ) = CyclotomicField.algebraBase _ _ _ :=
rfl
instance CyclotomicField.algebra' {R : Type*} [CommRing R] [Algebra R K] :
Algebra R (CyclotomicField n K) :=
SplittingField.algebra' (cyclotomic n K)
#align cyclotomic_field.algebra' CyclotomicField.algebra'
instance {R : Type*} [CommRing R] [Algebra R K] : IsScalarTower R K (CyclotomicField n K) :=
SplittingField.isScalarTower _
instance CyclotomicField.noZeroSMulDivisors : NoZeroSMulDivisors A (CyclotomicField n K) := by
refine' NoZeroSMulDivisors.of_algebraMap_injective _
rw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]
exact
(Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))
(IsFractionRing.injective A K) : _)
#align cyclotomic_field.no_zero_smul_divisors CyclotomicField.noZeroSMulDivisors
/-- If `A` is a domain with fraction field `K` and `n : ℕ+`, we define `CyclotomicRing n A K` as
the `A`-subalgebra of `CyclotomicField n K` generated by the roots of `X ^ n - 1`. If `n`
is nonzero in `A`, it has the instance `IsCyclotomicExtension {n} A (CyclotomicRing n A K)`. -/
@[nolint unusedArguments]
def CyclotomicRing : Type w :=
adjoin A {b : CyclotomicField n K | b ^ (n : ℕ) = 1}
--deriving CommRing, IsDomain, Inhabited
#align cyclotomic_ring CyclotomicRing
namespace CyclotomicRing
--Porting note: could not be derived
instance : CommRing (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : IsDomain (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
--Porting note: could not be derived
instance : Inhabited (CyclotomicRing n A K) := by
delta CyclotomicRing; infer_instance
/-- The `A`-algebra structure on `CyclotomicRing n A K`. -/
instance algebraBase : Algebra A (CyclotomicRing n A K) :=
(adjoin A _).algebra
#align cyclotomic_ring.algebra_base CyclotomicRing.algebraBase
-- Ensure that there is no diamonds with ℤ.
example {n : ℕ+} : CyclotomicRing.algebraBase n ℤ ℚ = algebraInt _ :=
rfl
instance : NoZeroSMulDivisors A (CyclotomicRing n A K) :=
(adjoin A _).noZeroSMulDivisors_bot
theorem algebraBase_injective : Function.Injective <| algebraMap A (CyclotomicRing n A K) :=
NoZeroSMulDivisors.algebraMap_injective _ _
#align cyclotomic_ring.algebra_base_injective CyclotomicRing.algebraBase_injective
instance : Algebra (CyclotomicRing n A K) (CyclotomicField n K) :=
(adjoin A _).toAlgebra
theorem adjoin_algebra_injective :
Function.Injective <| algebraMap (CyclotomicRing n A K) (CyclotomicField n K) :=
Subtype.val_injective
#align cyclotomic_ring.adjoin_algebra_injective CyclotomicRing.adjoin_algebra_injective
instance : NoZeroSMulDivisors (CyclotomicRing n A K) (CyclotomicField n K) :=
NoZeroSMulDivisors.of_algebraMap_injective (adjoin_algebra_injective n A K)
instance : IsScalarTower A (CyclotomicRing n A K) (CyclotomicField n K) :=
IsScalarTower.subalgebra' _ _ _ _
instance isCyclotomicExtension [NeZero ((n : ℕ) : A)] :
IsCyclotomicExtension {n} A (CyclotomicRing n A K) where
exists_prim_root := @fun a han => by
rw [mem_singleton_iff] at han
subst a
haveI := NeZero.of_noZeroSMulDivisors A K n
haveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n
obtain ⟨μ, hμ⟩ := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)
refine' ⟨⟨μ, subset_adjoin _⟩, _⟩
· apply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr
refine' ⟨n, Nat.mem_divisors_self _ n.ne_zero, _⟩
rwa [← isRoot_cyclotomic_iff] at hμ
· rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]
adjoin_roots x := by
refine'
adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x
· refine' subset_adjoin _
simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]
rwa [← Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]
· exact Subalgebra.algebraMap_mem _ a
· exact Subalgebra.add_mem _ hy hz
· exact Subalgebra.mul_mem _ hy hz
#align cyclotomic_ring.is_cyclotomic_extension CyclotomicRing.isCyclotomicExtension
instance [IsDomain A] [NeZero ((n : ℕ) : A)] :
IsFractionRing (CyclotomicRing n A K) (CyclotomicField n K) where
map_units' := fun ⟨x, hx⟩ => by
rw [isUnit_iff_ne_zero]
apply map_ne_zero_of_mem_nonZeroDivisors
apply adjoin_algebra_injective
exact hx
surj' x := by
letI : NeZero ((n : ℕ) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)
refine
Algebra.adjoin_induction
(((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1
(CyclotomicField.isCyclotomicExtension n K)).2
x)
(fun y hy => ?_) (fun k => ?_) ?_ ?_
-- Porting note: the last goal was `by simpa` that now fails.
· exact ⟨⟨⟨y, subset_adjoin hy⟩, 1⟩, by simp; rfl⟩
· have : IsLocalization (nonZeroDivisors A) K := inferInstance
replace := this.surj
obtain ⟨⟨z, w⟩, hw⟩ := this k
refine' ⟨⟨algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,
map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2⟩, _⟩
letI : IsScalarTower A K (CyclotomicField n K) :=
IsScalarTower.of_algebraMap_eq (congr_fun rfl)
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply,
@IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w,
← RingHom.map_mul, hw, ← IsScalarTower.algebraMap_apply]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, add_mul, ← mul_assoc, ha,
mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, hb]
simp only [map_add, map_mul]
· rintro y z ⟨a, ha⟩ ⟨b, hb⟩
refine' ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, _⟩
rw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), mul_assoc, ←
mul_assoc z, hb, ← mul_comm ((algebraMap (CyclotomicRing n A K) _) ↑a.2), ← mul_assoc, ha]
simp only [map_mul]
exists_of_eq {x y} h := ⟨1, by rw [adjoin_algebra_injective n A K h]⟩
theorem eq_adjoin_primitive_root {μ : CyclotomicField n K} (h : IsPrimitiveRoot μ n) :
CyclotomicRing n A K = adjoin A ({μ} : Set (CyclotomicField n K)) := by
rw [← IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]
simp [CyclotomicRing]
#align cyclotomic_ring.eq_adjoin_primitive_root CyclotomicRing.eq_adjoin_primitive_root
end CyclotomicRing
end CyclotomicRing
end IsDomain
section IsAlgClosed
variable [IsAlgClosed K]
/-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
refine' ⟨r, _⟩
haveI := h a ha
| rwa [coe_aeval_eq_eval, ← IsRoot.def, isRoot_cyclotomic_iff] at hr | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K := by
refine' ⟨@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _⟩
obtain ⟨r, hr⟩ := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'
refine' ⟨r, _⟩
haveI := h a ha
| Mathlib.NumberTheory.Cyclotomic.Basic.729_0.xReI1DeVvechFQU | /-- Algebraically closed fields are `S`-cyclotomic extensions over themselves if
`NeZero ((a : ℕ) : K))` for all `a ∈ S`. -/
theorem IsAlgClosed.isCyclotomicExtension (h : ∀ a ∈ S, NeZero ((a : ℕ) : K)) :
IsCyclotomicExtension S K K | Mathlib_NumberTheory_Cyclotomic_Basic |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ¬AffineIndependent 𝕜 f
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
| rw [affineIndependent_iff] at h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ¬∀ (s : Finset ι) (w : ι → 𝕜), Finset.sum s w = 0 → ∑ e in s, w e • f e = 0 → ∀ e ∈ s, w e = 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
| push_neg at h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
h : ∃ s w, Finset.sum s w = 0 ∧ ∑ e in s, w e • f e = 0 ∧ ∃ e ∈ s, w e ≠ 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
| obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
| let I : Finset ι := s.filter fun i ↦ 0 ≤ w i | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
| let J : Finset ι := s.filter fun i ↦ w i < 0 | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
| let p : E := centerMass I w f | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
| have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
⊢ ∑ j in J, w j + ∑ i in I, w i = 0 | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
| simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
| have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
⊢ 0 < ∑ i in I, w i | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
| rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
pos_w_index : ι
h1' : pos_w_index ∈ s
h2' : 0 < w pos_w_index
⊢ 0 < ∑ i in I, w i | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
| exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
pos_w_index : ι
h1' : pos_w_index ∈ s
h2' : 0 < w pos_w_index
⊢ pos_w_index ∈ I ∧ 0 < w pos_w_index | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by | simp only [mem_filter, h1', h2'.le, and_self, h2'] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
| have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
⊢ ∑ i in J, w i • f i + ∑ i in I, w i • f i = 0 | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
| simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
⊢ ∃ I, Set.Nonempty ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
| refine ⟨I, p, ?_, ?_⟩ | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
⊢ p ∈ (convexHull 𝕜) (f '' ↑I) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· | exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi) | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
⊢ p ∈ (convexHull 𝕜) (f '' (↑I)ᶜ) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
| rw [← hp] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
⊢ centerMass J w f ∈ (convexHull 𝕜) (f '' (↑I)ᶜ) | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
| refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_) | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
| Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2.refine_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
⊢ ∑ i in J, w i < 0 | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_)
· | linarith only [hI, hJI] | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_)
· | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
case intro.intro.intro.intro.intro.intro.refine_2.refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : Finset.sum s w = 0
h_vsum : ∑ e in s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := filter (fun i => 0 ≤ w i) s
J : Finset ι := filter (fun i => w i < 0) s
p : E := centerMass I w f
hJI : ∑ j in J, w j + ∑ i in I, w i = 0
hI : 0 < ∑ i in I, w i
hp : centerMass J w f = p
_i : ι
hi : _i ∈ J
hi' : _i ∈ ↑I
⊢ False | /-
Copyright (c) 2023 Vasily Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasily Nesterov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
/-!
# Radon's theorem on convex sets
Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex
hulls intersect.
## Tags
convex hull, radon, affine independence
-/
open Finset Set
open BigOperators
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
/-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_)
· linarith only [hI, hJI]
· | exact (mem_filter.mp hi').2.not_lt (mem_filter.mp hi).2 | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j in J, w j + ∑ i in I, w i = 0 := by
simpa only [h_wsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) w
have hI : 0 < ∑ i in I, w i := by
rcases exists_pos_of_sum_zero_of_exists_nonzero _ h_wsum ⟨nonzero_w_index, h1, h2⟩
with ⟨pos_w_index, h1', h2'⟩
exact sum_pos' (λ _i hi ↦ (mem_filter.1 hi).2)
⟨pos_w_index, by simp only [mem_filter, h1', h2'.le, and_self, h2']⟩
have hp : centerMass J w f = p := Finset.centerMass_of_sum_add_sum_eq_zero hJI $ by
simpa only [← h_vsum, not_lt] using sum_filter_add_sum_filter_not s (fun i ↦ w i < 0) _
refine ⟨I, p, ?_, ?_⟩
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
(fun _i hi ↦ Set.mem_image_of_mem _ hi)
rw [← hp]
refine centerMass_mem_convexHull_of_nonpos _ (fun _ hi ↦ (mem_filter.mp hi).2.le) ?_
(fun _i hi ↦ Set.mem_image_of_mem _ fun hi' ↦ ?_)
· linarith only [hI, hJI]
· | Mathlib.Analysis.Convex.Radon.25_0.TlRdL7CDP8NzLUh | /-- **Radon theorem on convex sets**: Any family `f` of affine dependent vectors contains a set `I`
with the property that convex hulls of `I` and `Iᶜ` intersect. -/
theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty | Mathlib_Analysis_Convex_Radon |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ : ℝ
⊢ taylorWithin f (n + 1) s x₀ =
taylorWithin f n s x₀ +
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀)) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
| dsimp only [taylorWithin] | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
| Mathlib.Analysis.Calculus.Taylor.75_0.INXnr4jrmq9RIjK | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ : ℝ
⊢ ∑ k in Finset.range (n + 1 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) =
∑ k in Finset.range (n + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀)) +
(PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀)) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
| rw [Finset.sum_range_succ] | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
| Mathlib.Analysis.Calculus.Taylor.75_0.INXnr4jrmq9RIjK | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ taylorWithinEval f (n + 1) s x₀ x =
taylorWithinEval f n s x₀ x + (((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
| simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval x) (taylorWithin f n s x₀) +
(PolynomialModule.eval (Polynomial.eval x (Polynomial.X - Polynomial.C x₀)))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀)) =
(PolynomialModule.eval x) (taylorWithin f n s x₀) +
(((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
| congr | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (PolynomialModule.eval (Polynomial.eval x (Polynomial.X - Polynomial.C x₀)))
((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀)) =
(((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
| simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |
case e_a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (x - x₀) ^ (n + 1) • taylorCoeffWithin f (n + 1) s x₀ =
((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : ℝ → E`,
where `E` is a normed vector space over `ℝ` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {𝕜 E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E :=
(k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/
noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E :=
PolynomialModule.eval x (taylorWithin f n s x₀)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
| dsimp only [taylorCoeffWithin] | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ | Mathlib_Analysis_Calculus_Taylor |