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
• IsGalois.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.

Stacks Tag 09I0

• to_isSeparable : Algebra.IsSeparable F E
• to_normal : 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

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 (↥(FixedPoints.subfield G E)) E

Let $E$ be a field. Let $G$ be a finite group acting on $E$. Then the extension $E / E^G$ is Galois.

Stacks Tag 09I3 (first part)

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 ↥F⟮α⟯) (minpoly F α)) : Fintype.card (↥F⟮α⟯ ≃ₐ[F] ↥F⟮α⟯) = Module.finrank F ↥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) = Module.finrank F E

Let $E / F$ be a finite extension of fields. If $E$ is Galois over $F$, then $|\text{Aut}(E/F)| = [E : F]$.

Stacks Tag 09I1 ('only if' part)

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

Let $E / K / F$ be a tower of field extensions. If $E$ is Galois over $F$, then $E$ is Galois over $K$.

Stacks Tag 09I2

@[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 (↥K) E

theorem isGalois_iff_isGalois_bot {F : Type u_1} {E : Type u_3} [Field F] [Field E] [Algebra F E] : IsGalois (↥⊥) 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 ↥⊤ ↔ IsGalois F E

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

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 ↥H

theorem IntermediateField.mem_fixedField_iff {F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (H : Subgroup (E ≃ₐ[F] E)) (x : E) : x ∈ IntermediateField.fixedField H ↔ ∀ f ∈ H, f x = x

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] : Module.finrank (↥(IntermediateField.fixedField H)) E = Fintype.card ↥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) : ↥K.fixingSubgroup ≃* E ≃ₐ[↥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 ↥K ↥(IntermediateField.fixedField K.fixingSubgroup)

Equations

• IntermediateField.fixedField.smul K = { smul := fun (x : ↥K) (y : ↥(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 ↥K ↥(IntermediateField.fixedField K.fixingSubgroup)

Equations

• IntermediateField.fixedField.algebra K = Algebra.mk { toFun := fun (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 (↥K) (↥(IntermediateField.fixedField K.fixingSubgroup)) E

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 ↥K.fixingSubgroup = Module.finrank (↥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.

Stacks Tag 09DW

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 IntermediateField.restrictNormalHom_ker {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (E : IntermediateField K L) [Normal K ↥E] : (AlgEquiv.restrictNormalHom ↥E).ker = E.fixingSubgroup

instance IsGalois.of_fixedField_normal_subgroup {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] [IsGalois K L] (H : Subgroup (L ≃ₐ[K] L)) [hn : H.Normal] : IsGalois K ↥(IntermediateField.fixedField H)

If H is a normal Subgroup of Gal(L / K), then fixedField H is Galois over K.

noncomputable def IsGalois.normalAutEquivQuotient {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] (H : Subgroup (L ≃ₐ[K] L)) [H.Normal] : (L ≃ₐ[K] L) ⧸ H ≃* ↥(IntermediateField.fixedField H) ≃ₐ[K] ↥(IntermediateField.fixedField H)

If H is a normal Subgroup of Gal(L / K), then Gal(fixedField H / K) is isomorphic to Gal(L / K) ⧸ H.

Equations

• IsGalois.normalAutEquivQuotient H = (QuotientGroup.quotientMulEquivOfEq ⋯).trans (QuotientGroup.quotientKerEquivOfSurjective (AlgEquiv.restrictNormalHom ↥(IntermediateField.fixedField H)) ⋯)

theorem IsGalois.normalAutEquivQuotient_apply {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] (H : Subgroup (L ≃ₐ[K] L)) [H.Normal] (σ : L ≃ₐ[K] L) : (IsGalois.normalAutEquivQuotient H) ↑σ = (AlgEquiv.restrictNormalHom ↥(IntermediateField.fixedField H)) σ

@[simp]

theorem IsGalois.map_fixingSubgroup {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (E : IntermediateField K L) (σ : L ≃ₐ[K] L) : (IntermediateField.map (↑σ) E).fixingSubgroup = MulAut.conj σ • E.fixingSubgroup

instance IsGalois.fixingSubgroup_normal_of_isGalois {K : Type u_3} {L : Type u_4} [Field K] [Field L] [Algebra K L] (E : IntermediateField K L) [IsGalois K L] [IsGalois K ↥E] : E.fixingSubgroup.Normal

Let E be an intermediateField of a Galois extension L / K. If E / K is Galois extension, then E.fixingSubgroup is a normal subgroup of Gal(L / K).

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) = Module.finrank F E) : IsGalois F E

Let $E / F$ be a finite extension of fields. If $|\text{Aut}(E/F)| = [E : F]$, then $E$ is Galois over $F$.

Stacks Tag 09I1 ('if' part)

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 (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)] : Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) = Fintype.card (K →ₐ[F] E) * Module.finrank K ↥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) = Module.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 ↥(normalClosure k K F)

Let $F / K / k$ be a tower of field extensions. If $F$ is Galois over $k$, then the normal closure of $K$ over $k$ in $F$ is Galois over $k$.

Stacks Tag 0EXM

@[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