Galois Extensions

In this file we define Galois extensions as extensions which are both separable and normal.

Main definitions

• IsGalois F E where E is an extension of F
• fixedField H where H : Subgroup (E ≃ₐ[F] E)
• fixingSubgroup K where K : IntermediateField F E
• intermediateFieldEquivSubgroup where E/F is finite dimensional and Galois

Main results

• IntermediateField.fixingSubgroup_fixedField : If E/F is finite dimensional (but not necessarily Galois) then fixingSubgroup (fixedField H) = H
• IntermediateField.fixedField_fixingSubgroup: If E/F is finite dimensional and Galois then fixedField (fixingSubgroup K) = K

Together, these two results prove the Galois correspondence.

• IsGalois.tfae : Equivalent characterizations of a Galois extension of finite degree

class IsGalois (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] : Prop

A field extension E/F is Galois if it is both separable and normal. Note that in mathlib a separable extension of fields is by definition algebraic.

• to_isSeparable : Algebra.IsSeparable F E
• to_normal : Normal F E

theorem IsGalois.to_isSeparable {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [self : IsGalois F E] : Algebra.IsSeparable F E

theorem IsGalois.to_normal {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [self : IsGalois F E] : Normal F E

theorem isGalois_iff {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] : IsGalois F E ↔ Algebra.IsSeparable F E ∧ Normal F E

instance IsGalois.self (F : Type u_1) [Field F] : IsGalois F F

Equations

• ⋯ = ⋯

theorem IsGalois.integral (F : Type u_1) [Field F] {E : Type u_2} [Field E] [Algebra F E] [IsGalois F E] (x : E) : IsIntegral F x

theorem IsGalois.separable (F : Type u_1) [Field F] {E : Type u_2} [Field E] [Algebra F E] [IsGalois F E] (x : E) : IsSeparable F x

theorem IsGalois.splits (F : Type u_1) [Field F] {E : Type u_2} [Field E] [Algebra F E] [IsGalois F E] (x : E) : Polynomial.Splits (algebraMap F E) (minpoly F x)

instance IsGalois.of_fixed_field (E : Type u_2) [Field E] (G : Type u_3) [Group G] [Finite G] [MulSemiringAction G E] : IsGalois { x : E // x ∈ FixedPoints.subfield G E } E

Equations

• ⋯ = ⋯

theorem IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrank (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [FiniteDimensional F E] {α : E} (hα : IsIntegral F α) (h_sep : IsSeparable F α) (h_splits : Polynomial.Splits (algebraMap F { x : E // x ∈ F⟮α⟯ }) (minpoly F α)) : Fintype.card ({ x : E // x ∈ F⟮α⟯ } ≃ₐ[F] { x : E // x ∈ F⟮α⟯ }) = FiniteDimensional.finrank F { x : E // x ∈ F⟮α⟯ }

theorem IsGalois.card_aut_eq_finrank (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [FiniteDimensional F E] [IsGalois F E] : Fintype.card (E ≃ₐ[F] E) = FiniteDimensional.finrank F E

theorem IsGalois.tower_top_of_isGalois (F : Type u_1) (K : Type u_2) (E : Type u_3) [Field F] [Field K] [Field E] [Algebra F K] [Algebra F E] [Algebra K E] [IsScalarTower F K E] [IsGalois F E] : IsGalois K E

@[instance 100]

instance IsGalois.tower_top_intermediateField {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] (K : IntermediateField F E) [IsGalois F E] : IsGalois { x : E // x ∈ K } E

Equations

• ⋯ = ⋯

theorem isGalois_iff_isGalois_bot {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] : IsGalois { x : E // x ∈ ⊥ } E ↔ IsGalois F E

theorem IsGalois.of_algEquiv {F : Type u_1} {E : Type u_3} [Field F] [Field E] {E' : Type u_4} [Field E'] [Algebra F E'] [Algebra F E] [IsGalois F E] (f : E ≃ₐ[F] E') : IsGalois F E'

theorem AlgEquiv.transfer_galois {F : Type u_1} {E : Type u_3} [Field F] [Field E] {E' : Type u_4} [Field E'] [Algebra F E'] [Algebra F E] (f : E ≃ₐ[F] E') : IsGalois F E ↔ IsGalois F E'

theorem isGalois_iff_isGalois_top {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] : IsGalois F { x : E // x ∈ ⊤ } ↔ IsGalois F E

instance isGalois_bot {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] : IsGalois F { x : E // x ∈ ⊥ }

Equations

• ⋯ = ⋯

def FixedPoints.intermediateField {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (M : Type u_3) [Monoid M] [MulSemiringAction M E] [SMulCommClass M F E] : IntermediateField F E

The intermediate field of fixed points fixed by a monoid action that commutes with the F-action on E.

Equations

• FixedPoints.intermediateField M = { carrier := MulAction.fixedPoints M E, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯, inv_mem' := ⋯ }

def IntermediateField.fixedField {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (H : Subgroup (E ≃ₐ[F] E)) : IntermediateField F E

The intermediate field fixed by a subgroup

Equations

• IntermediateField.fixedField H = FixedPoints.intermediateField { x : E ≃ₐ[F] E // x ∈ H }

theorem IntermediateField.finrank_fixedField_eq_card {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (H : Subgroup (E ≃ₐ[F] E)) [FiniteDimensional F E] [DecidablePred fun (x : E ≃ₐ[F] E) => x ∈ H] : FiniteDimensional.finrank { x : E // x ∈ IntermediateField.fixedField H } E = Fintype.card { x : E ≃ₐ[F] E // x ∈ H }

def IntermediateField.fixingSubgroup {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) : Subgroup (E ≃ₐ[F] E)

The subgroup fixing an intermediate field

Equations

• K.fixingSubgroup = fixingSubgroup (E ≃ₐ[F] E) ↑K

theorem IntermediateField.le_iff_le {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (H : Subgroup (E ≃ₐ[F] E)) (K : IntermediateField F E) : K ≤ IntermediateField.fixedField H ↔ H ≤ K.fixingSubgroup

def IntermediateField.fixingSubgroupEquiv {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) : { x : E ≃ₐ[F] E // x ∈ K.fixingSubgroup } ≃* E ≃ₐ[{ x : E // x ∈ K }] E

The fixing subgroup of K : IntermediateField F E is isomorphic to E ≃ₐ[K] E

Equations

• One or more equations did not get rendered due to their size.

theorem IntermediateField.fixingSubgroup_fixedField {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (H : Subgroup (E ≃ₐ[F] E)) [FiniteDimensional F E] : (IntermediateField.fixedField H).fixingSubgroup = H

instance IntermediateField.fixedField.smul {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) : SMul { x : E // x ∈ K } { x : E // x ∈ IntermediateField.fixedField K.fixingSubgroup }

Equations

• IntermediateField.fixedField.smul K = { smul := fun (x : { x : E // x ∈ K }) (y : { x : E // x ∈ IntermediateField.fixedField K.fixingSubgroup }) => ⟨↑x * ↑y, ⋯⟩ }

instance IntermediateField.fixedField.algebra {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) : Algebra { x : E // x ∈ K } { x : E // x ∈ IntermediateField.fixedField K.fixingSubgroup }

Equations

• IntermediateField.fixedField.algebra K = Algebra.mk { toFun := fun (x : { x : E // x ∈ K }) => ⟨↑x, ⋯⟩, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯

instance IntermediateField.fixedField.isScalarTower {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) : IsScalarTower { x : E // x ∈ K } { x : E // x ∈ IntermediateField.fixedField K.fixingSubgroup } E

Equations

• ⋯ = ⋯

theorem IsGalois.fixedField_fixingSubgroup {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) [FiniteDimensional F E] [h : IsGalois F E] : IntermediateField.fixedField K.fixingSubgroup = K

theorem IsGalois.card_fixingSubgroup_eq_finrank {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (K : IntermediateField F E) [DecidablePred fun (x : E ≃ₐ[F] E) => x ∈ K.fixingSubgroup] [FiniteDimensional F E] [IsGalois F E] : Fintype.card { x : E ≃ₐ[F] E // x ∈ K.fixingSubgroup } = FiniteDimensional.finrank { x : E // x ∈ K } E

def IsGalois.intermediateFieldEquivSubgroup {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [FiniteDimensional F E] [IsGalois F E] : IntermediateField F E ≃o (Subgroup (E ≃ₐ[F] E))ᵒᵈ

The Galois correspondence from intermediate fields to subgroups

Equations

• IsGalois.intermediateFieldEquivSubgroup = { toFun := IntermediateField.fixingSubgroup, invFun := IntermediateField.fixedField, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }

def IsGalois.galoisInsertionIntermediateFieldSubgroup {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [FiniteDimensional F E] : GaloisInsertion (⇑OrderDual.toDual ∘ IntermediateField.fixingSubgroup) (IntermediateField.fixedField ∘ ⇑OrderDual.toDual)

The Galois correspondence as a GaloisInsertion

Equations

• One or more equations did not get rendered due to their size.

def IsGalois.galoisCoinsertionIntermediateFieldSubgroup {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [FiniteDimensional F E] [IsGalois F E] : GaloisCoinsertion (⇑OrderDual.toDual ∘ IntermediateField.fixingSubgroup) (IntermediateField.fixedField ∘ ⇑OrderDual.toDual)

The Galois correspondence as a GaloisCoinsertion

Equations

• One or more equations did not get rendered due to their size.

theorem IsGalois.is_separable_splitting_field (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [FiniteDimensional F E] [IsGalois F E] : ∃ (p : Polynomial F), p.Separable ∧ Polynomial.IsSplittingField F E p

theorem IsGalois.of_fixedField_eq_bot (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [FiniteDimensional F E] (h : IntermediateField.fixedField ⊤ = ⊥) : IsGalois F E

theorem IsGalois.of_card_aut_eq_finrank (F : Type u_1) [Field F] (E : Type u_2) [Field E] [Algebra F E] [FiniteDimensional F E] (h : Fintype.card (E ≃ₐ[F] E) = FiniteDimensional.finrank F E) : IsGalois F E

theorem IsGalois.of_separable_splitting_field_aux {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] {p : Polynomial F} [hFE : FiniteDimensional F E] [sp : Polynomial.IsSplittingField F E p] (hp : p.Separable) (K : Type u_3) [Field K] [Algebra F K] [Algebra K E] [IsScalarTower F K E] {x : E} (hx : x ∈ p.aroots E) [Fintype (K →ₐ[F] E)] [Fintype ({ x_1 : E // x_1 ∈ IntermediateField.restrictScalars F K⟮x⟯ } →ₐ[F] E)] : Fintype.card ({ x_1 : E // x_1 ∈ IntermediateField.restrictScalars F K⟮x⟯ } →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * FiniteDimensional.finrank K { x_1 : E // x_1 ∈ K⟮x⟯ }

theorem IsGalois.of_separable_splitting_field {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] {p : Polynomial F} [sp : Polynomial.IsSplittingField F E p] (hp : p.Separable) : IsGalois F E

theorem IsGalois.tfae {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] [FiniteDimensional F E] : [IsGalois F E, IntermediateField.fixedField ⊤ = ⊥, Fintype.card (E ≃ₐ[F] E) = FiniteDimensional.finrank F E, ∃ (p : Polynomial F), p.Separable ∧ Polynomial.IsSplittingField F E p].TFAE

Equivalent characterizations of a Galois extension of finite degree

instance IsGalois.normalClosure (k : Type u_1) (K : Type u_2) (F : Type u_3) [Field k] [Field K] [Field F] [Algebra k K] [Algebra k F] [IsGalois k F] : IsGalois k { x : F // x ∈ normalClosure k K F }

Equations

• ⋯ = ⋯

@[instance 100]

instance IsAlgClosure.isGalois (k : Type u_1) (K : Type u_2) [Field k] [Field K] [Algebra k K] [IsAlgClosure k K] [CharZero k] : IsGalois k K

Equations

• ⋯ = ⋯