Predicate for Galois Groups

Given an action of a group G on an extension of fields L/K, we introduce a predicate IsGaloisGroup G K L saying that G acts faithfully on L with fixed field K. In particular, we do not assume that L is an algebraic extension of K.

Implementation notes

We actually define IsGaloisGroup G A B for extensions of rings B/A, with the same definition (faithful action on B with fixed ring A). This definition turns out to axiomatize a common setup in algebraic number theory where a Galois group Gal(L/K) acts on an extension of subrings B/A (e.g., rings of integers). In particular, there are theorems in algebraic number theory that naturally assume [IsGaloisGroup G A B] and whose statements would otherwise require assuming (K L : Type*) [Field K] [Field L] [Algebra K L] [IsGalois K L] (along with predicates relating K and L to the rings A and B) despite K and L not appearing in the conclusion.

Unfortunately, this definition of IsGaloisGroup G A B for extensions of rings B/A is nonstandard and clashes with other notions such as the étale fundamental group. In particular, if G is finite and A is integrally closed, then IsGaloisGroup G A B is equivalent to B/A being integral and the fields of fractions Frac(B)/Frac(A) being Galois with Galois group G (see IsGaloisGroup.iff_isFractionRing), rather than B/A being étale for instance.

But in the absence of a more suitable name, the utility of the predicate IsGaloisGroup G A B for extensions of rings B/A seems to outweigh these terminological issues.

class IsGaloisGroup (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] : Prop

G is a Galois group for L/K if the action of G on L is faithful with fixed field K. In particular, we do not assume that L is an algebraic extension of K.

See the implementation notes in this file for the meaning of this definition in the case of rings.

• faithful : FaithfulSMul G B
• commutes : SMulCommClass G A B
• isInvariant : Algebra.IsInvariant A B G

theorem IsGaloisGroup.of_mulEquiv {G : Type u_1} {A : Type u_2} {B : Type u_4} [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hG : IsGaloisGroup G A B] {H : Type u_5} [Group H] [MulSemiringAction H B] (e : H ≃* G) (he : ∀ (h : H) (x : B), e h • x = h • x) : IsGaloisGroup H A B

theorem IsGaloisGroup.iff_of_mulEquiv {G : Type u_1} {A : Type u_2} {B : Type u_4} [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] {H : Type u_5} [Group H] [MulSemiringAction H B] (e : H ≃* G) (he : ∀ (h : H) (x : B), e h • x = h • x) : IsGaloisGroup H A B ↔ IsGaloisGroup G A B

@[simp]

theorem IsGaloisGroup.top_iff {G : Type u_1} {A : Type u_2} {B : Type u_4} [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] : IsGaloisGroup (↥⊤) A B ↔ IsGaloisGroup G A B

instance instIsGaloisGroupSubtypeMemSubgroupTop (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [IsGaloisGroup G A B] : IsGaloisGroup (↥⊤) A B

theorem IsGaloisGroup.of_algEquiv (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hG : IsGaloisGroup G A B] (B' : Type u_5) [Semiring B'] [Algebra A B'] [MulSemiringAction G B'] (e : B ≃ₐ[A] B') (he : ∀ (g : G) (x : B), e (g • x) = g • e x) : IsGaloisGroup G A B'

theorem IsGaloisGroup.of_ringEquiv (G : Type u_1) (A : Type u_2) (A' : Type u_3) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hG : IsGaloisGroup G A B] [CommSemiring A'] [Algebra A' B] (e : A ≃+* A') (he : ∀ (a : A), (algebraMap A' B) (e a) = (algebraMap A B) a) : IsGaloisGroup G A' B

theorem Subgroup.smul_algebraMap {G : Type u_1} (B : Type u_4) [Group G] [Semiring B] [MulSemiringAction G B] {C : Type u_5} [CommSemiring C] [Algebra C B] {H : Subgroup G} [SMulCommClass (↥H) C B] {g : G} (hg : g ∈ H) (x : C) : g • (algebraMap C B) x = (algebraMap C B) x

theorem IsGaloisGroup.smul_mem_of_normal (G : Type u_1) (B : Type u_4) [Group G] [Semiring B] [MulSemiringAction G B] {C : Type u_5} [CommSemiring C] [Algebra C B] (N : Subgroup G) [hN : N.Normal] [hC : IsGaloisGroup (↥N) C B] (g : G) (x : C) : g • (algebraMap C B) x ∈ Set.range ⇑(algebraMap C B)

@[deprecated Subgroup.smul_algebraMap (since := "2026-05-28")]

theorem smul_eq_self {G : Type u_1} (B : Type u_4) [Group G] [Semiring B] [MulSemiringAction G B] {C : Type u_5} [CommSemiring C] [Algebra C B] {H : Subgroup G} [SMulCommClass (↥H) C B] {g : G} (hg : g ∈ H) (x : C) : g • (algebraMap C B) x = (algebraMap C B) x

Alias of Subgroup.smul_algebraMap.

@[deprecated IsGaloisGroup.smul_mem_of_normal (since := "2026-05-28")]

theorem smul_mem_of_normal (G : Type u_1) (B : Type u_4) [Group G] [Semiring B] [MulSemiringAction G B] {C : Type u_5} [CommSemiring C] [Algebra C B] (N : Subgroup G) [hN : N.Normal] [hC : IsGaloisGroup (↥N) C B] (g : G) (x : C) : g • (algebraMap C B) x ∈ Set.range ⇑(algebraMap C B)

Alias of IsGaloisGroup.smul_mem_of_normal.

noncomputable def IsGaloisGroup.ringEquivFixedPoints (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] : A ≃+* ↥(FixedPoints.subsemiring B G)

If B/A is Galois with Galois group G, then A is isomorphic to the subring of elements of B fixed by G.

Equations

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

@[simp]

theorem IsGaloisGroup.ringEquivFixedPoints_apply_coe (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] (x : A) : ↑((ringEquivFixedPoints G A B) x) = (algebraMap A B) x

@[simp]

theorem IsGaloisGroup.algebraMap_ringEquivFixedPoints_symm_apply (G : Type u_1) (A : Type u_2) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] (x : ↥(FixedPoints.subsemiring B G)) : (algebraMap A B) ((ringEquivFixedPoints G A B).symm x) = ↑x

noncomputable def IsGaloisGroup.ringEquiv (G : Type u_1) (A : Type u_2) (A' : Type u_3) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] [CommSemiring A'] [Algebra A' B] [FaithfulSMul A' B] [hA' : IsGaloisGroup G A' B] : A ≃+* A'

If B/A and B/A' are Galois with the same Galois group, then A ≃+* A'.

Equations

• IsGaloisGroup.ringEquiv G A A' B = (IsGaloisGroup.ringEquivFixedPoints G A B).trans (IsGaloisGroup.ringEquivFixedPoints G A' B).symm

@[simp]

theorem IsGaloisGroup.algebraMap_ringEquiv_apply (G : Type u_1) (A : Type u_2) (A' : Type u_3) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] [CommSemiring A'] [Algebra A' B] [FaithfulSMul A' B] [hA' : IsGaloisGroup G A' B] (x : A) : (algebraMap A' B) ((ringEquiv G A A' B) x) = (algebraMap A B) x

@[simp]

theorem IsGaloisGroup.algebraMap_ringEquiv_symm_apply (G : Type u_1) (A : Type u_2) (A' : Type u_3) (B : Type u_4) [Group G] [CommSemiring A] [Semiring B] [Algebra A B] [MulSemiringAction G B] [hA : IsGaloisGroup G A B] [FaithfulSMul A B] [CommSemiring A'] [Algebra A' B] [FaithfulSMul A' B] [hA' : IsGaloisGroup G A' B] (x : A') : (algebraMap A B) ((ringEquiv G A A' B).symm x) = (algebraMap A' B) x

theorem IsGaloisGroup.to_isFractionRing_of_isIntegral (G : Type u_1) (A : Type u_2) (B : Type u_3) (K : Type u_4) (L : Type u_5) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [Field K] [Field L] [Algebra K L] [Algebra A K] [Algebra B L] [Algebra A L] [IsFractionRing A K] [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [MulSemiringAction G L] [SMulDistribClass G B L] [Algebra.IsIntegral A B] [hGAB : IsGaloisGroup G A B] : IsGaloisGroup G K L

IsGaloisGroup for rings implies IsGaloisGroup for their fraction fields.

theorem IsGaloisGroup.to_isFractionRing (G : Type u_1) (A : Type u_2) (B : Type u_3) (K : Type u_4) (L : Type u_5) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [Field K] [Field L] [Algebra K L] [Algebra A K] [Algebra B L] [Algebra A L] [IsFractionRing A K] [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [MulSemiringAction G L] [SMulDistribClass G B L] [Finite G] [hGAB : IsGaloisGroup G A B] : IsGaloisGroup G K L

IsGaloisGroup for rings implies IsGaloisGroup for their fraction fields.

theorem IsGaloisGroup.of_isFractionRing (G : Type u_1) (A : Type u_2) (B : Type u_3) (K : Type u_4) (L : Type u_5) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [Field K] [Field L] [Algebra K L] [Algebra A K] [Algebra B L] [Algebra A L] [IsFractionRing A K] [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [MulSemiringAction G L] [SMulDistribClass G B L] [hGKL : IsGaloisGroup G K L] [IsIntegrallyClosed A] [Algebra.IsIntegral A B] : IsGaloisGroup G A B

If B is an integral extension of an integrally closed domain A, then IsGaloisGroup for their fraction fields implies IsGaloisGroup for these rings.

theorem IsGaloisGroup.iff_isFractionRing (G : Type u_1) (A : Type u_2) (B : Type u_3) (K : Type u_4) (L : Type u_5) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [Field K] [Field L] [Algebra K L] [Algebra A K] [Algebra B L] [Algebra A L] [IsFractionRing A K] [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [MulSemiringAction G L] [SMulDistribClass G B L] [Finite G] [IsIntegrallyClosed A] : IsGaloisGroup G A B ↔ Algebra.IsIntegral A B ∧ IsGaloisGroup G K L

If G is finite and A is integrally closed then IsGaloisGroup G A B is equivalent to B/A being integral and the fields of fractions Frac(B)/Frac(A) being Galois with Galois group G.

@[deprecated IsFractionRing.mulSemiringAction (since := "2026-04-20")]

def FractionRing.mulSemiringAction_of_isGaloisGroup (G : Type u_10) (A : Type u_11) (B : Type u_12) (K : Type u_13) (L : Type u_14) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [Field K] [Field L] [Algebra K L] [Algebra A K] [Algebra B L] [Algebra A L] [IsFractionRing A K] [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [SMulCommClass G A B] : MulSemiringAction G L

Alias of IsFractionRing.mulSemiringAction.

Equations

• FractionRing.mulSemiringAction_of_isGaloisGroup = IsFractionRing.mulSemiringAction

theorem IsGaloisGroup.toFractionRing (G : Type u_1) (A : Type u_2) (B : Type u_3) [Group G] [CommRing A] [CommRing B] [MulSemiringAction G B] [Algebra A B] [IsDomain A] [IsDomain B] [Module.IsTorsionFree A B] [Finite G] [IsGaloisGroup G A B] : IsGaloisGroup G (FractionRing A) (FractionRing B)

If G is finite and IsGaloisGroup G A B with A and B domains, then G is also a Galois group for FractionRing B / FractionRing A for the action defined by IsFractionRing.mulSemiringAction.

instance instIsGaloisGroupRingOfIntegersOfNumberField (K : Type u_6) (L : Type u_7) [Field K] [Field L] [NumberField K] [NumberField L] [Algebra K L] (G : Type u_8) [Group G] [MulSemiringAction G L] [IsGaloisGroup G K L] : IsGaloisGroup G (NumberField.RingOfIntegers K) (NumberField.RingOfIntegers L)

instance instIsGaloisGroupIntRingOfIntegersOfRat (L : Type u_6) [Field L] [NumberField L] (G : Type u_7) [Group G] [MulSemiringAction G L] [IsGaloisGroup G ℚ L] : IsGaloisGroup G ℤ (NumberField.RingOfIntegers L)

theorem IsGaloisGroup.fixedPoints_eq_bot (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] : FixedPoints.intermediateField G = ⊥

theorem IsGaloisGroup.isGalois (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [Finite G] [IsGaloisGroup G K L] : IsGalois K L

If G is a finite Galois group for L/K, then L/K is a Galois extension.

instance IsGaloisGroup.of_isGalois (K : Type u_3) (L : Type u_4) [Field K] [Field L] [Algebra K L] [IsGalois K L] : IsGaloisGroup Gal(L/K) K L

If L/K is a Galois extension, then Gal(L/K) is a Galois group for L/K.

theorem IsGaloisGroup.card_eq_finrank (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] : Nat.card G = Module.finrank K L

theorem IsGaloisGroup.finiteDimensional (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [Finite G] [IsGaloisGroup G K L] : FiniteDimensional K L

theorem IsGaloisGroup.finite (G : Type u_1) [Group G] (A : Type u_5) (B : Type u_6) [CommRing A] [CommRing B] [Algebra A B] [Module.Finite A B] [IsDomain A] [IsDomain B] [FaithfulSMul A B] [MulSemiringAction G B] [IsGaloisGroup G A B] : Finite G

noncomputable def IsGaloisGroup.mulEquivAlgEquiv (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] [Finite G] : G ≃* Gal(L/K)

If G is a finite Galois group for L/K, then G is isomorphic to Gal(L/K).

Equations

• IsGaloisGroup.mulEquivAlgEquiv G K L = MulEquiv.ofBijective (MulSemiringAction.toAlgAut G K L) ⋯

@[simp]

theorem IsGaloisGroup.mulEquivAlgEquiv_symm_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] [Finite G] (b : Gal(L/K)) : (mulEquivAlgEquiv G K L).symm b = Function.surjInv ⋯ b

@[simp]

theorem IsGaloisGroup.mulEquivAlgEquiv_apply_symm_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] [Finite G] (a : G) (a✝ : L) : ((mulEquivAlgEquiv G K L) a).symm a✝ = a⁻¹ • a✝

@[simp]

theorem IsGaloisGroup.mulEquivAlgEquiv_apply_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] [Finite G] (a : G) (a✝ : L) : ((mulEquivAlgEquiv G K L) a) a✝ = a • a✝

@[simp]

theorem IsGaloisGroup.map_mulEquivAlgEquiv_fixingSubgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGaloisGroup G K L] [Finite G] (F : IntermediateField K L) : Subgroup.map (↑(mulEquivAlgEquiv G K L)) (fixingSubgroup G ↑F) = F.fixingSubgroup

noncomputable def IsGaloisGroup.mulEquivCongr' (G : Type u_1) (G' : Type u_2) (K : Type u_3) (L : Type u_4) [Group G] [Group G'] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [MulSemiringAction G' L] [IsGaloisGroup G K L] [Finite G] [IsGaloisGroup G' K L] [Finite G'] : G ≃* G'

If G and G' are finite Galois groups for L/K, then G is isomorphic to G'. See mulEquivCongr for a more general version.

Equations

• IsGaloisGroup.mulEquivCongr' G G' K L = (IsGaloisGroup.mulEquivAlgEquiv G K L).trans (IsGaloisGroup.mulEquivAlgEquiv G' K L).symm

@[simp]

theorem IsGaloisGroup.mulEquivCongr'_apply_smul (G : Type u_1) (G' : Type u_2) (K : Type u_3) (L : Type u_4) [Group G] [Group G'] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [MulSemiringAction G' L] [IsGaloisGroup G K L] [Finite G] [IsGaloisGroup G' K L] [Finite G'] (g : G) (x : L) : (mulEquivCongr' G G' K L) g • x = g • x

noncomputable def IsGaloisGroup.mulEquivCongr (G : Type u_1) (G' : Type u_2) [Group G] [Group G'] [Finite G] [Finite G'] (A : Type u_5) (B : Type u_6) [CommRing A] [CommRing B] [IsDomain B] [Algebra A B] [FaithfulSMul A B] [MulSemiringAction G B] [MulSemiringAction G' B] [IsGaloisGroup G A B] [IsGaloisGroup G' A B] : G ≃* G'

If G and G' are finite Galois groups for B/A with B a domain, then G is isomorphic to G'.

Equations

• IsGaloisGroup.mulEquivCongr G G' A B = IsGaloisGroup.mulEquivCongr' G G' (FractionRing A) (FractionRing B)

@[simp]

theorem IsGaloisGroup.mulEquivCongr_apply_smul (G : Type u_1) (G' : Type u_2) [Group G] [Group G'] [Finite G] [Finite G'] (A : Type u_5) (B : Type u_6) [CommRing A] [CommRing B] [IsDomain B] [Algebra A B] [FaithfulSMul A B] [MulSemiringAction G B] [MulSemiringAction G' B] [IsGaloisGroup G A B] [IsGaloisGroup G' A B] (g : G) (x : B) : (mulEquivCongr G G' A B) g • x = g • x

@[simp]

theorem IsGaloisGroup.mulEquivCongr_symm_apply_smul (G : Type u_1) (G' : Type u_2) [Group G] [Group G'] [Finite G] [Finite G'] (A : Type u_5) (B : Type u_6) [CommRing A] [CommRing B] [IsDomain B] [Algebra A B] [FaithfulSMul A B] [MulSemiringAction G B] [MulSemiringAction G' B] [IsGaloisGroup G A B] [IsGaloisGroup G' A B] (g : G') (x : B) : (mulEquivCongr G G' A B).symm g • x = g • x

instance IsGaloisGroup.instSubtypeMemSubgroupSubalgebraSubalgebra (G : Type u_1) [Group G] (H : Subgroup G) (R : Type u_5) (S : Type u_6) [CommRing R] [CommRing S] [Algebra R S] [MulSemiringAction G S] [hGKL : IsGaloisGroup G R S] : IsGaloisGroup (↥H) (↥(FixedPoints.subalgebra R S ↥H)) S

instance IsGaloisGroup.subgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) [hGKL : IsGaloisGroup G K L] : IsGaloisGroup (↥H) (↥(FixedPoints.intermediateField ↥H)) L

theorem IsGaloisGroup.fixedPoints_of_isGaloisGroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) (F : IntermediateField K L) [hGKL : IsGaloisGroup G K L] [hHFL : IsGaloisGroup (↥H) (↥F) L] : FixedPoints.intermediateField ↥H = F

theorem IsGaloisGroup.of_fixedPoints_eq (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) (F : IntermediateField K L) [hGKL : IsGaloisGroup G K L] (hF : FixedPoints.intermediateField ↥H = F) : IsGaloisGroup (↥H) (↥F) L

theorem IsGaloisGroup.subgroup_iff {G : Type u_1} {K : Type u_3} {L : Type u_4} [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] {H : Subgroup G} {F : IntermediateField K L} [hGKL : IsGaloisGroup G K L] : IsGaloisGroup (↥H) (↥F) L ↔ FixedPoints.intermediateField ↥H = F

@[simp]

theorem IsGaloisGroup.finrank_fixedPoints_eq_card_subgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) [IsGaloisGroup G K L] : Module.finrank (↥(FixedPoints.intermediateField ↥H)) L = Nat.card ↥H

theorem IsGaloisGroup.of_mulEquiv_algEquiv {G : Type u_1} {K : Type u_3} {L : Type u_4} [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [IsGalois K L] (e : G ≃* Gal(L/K)) (he : ∀ (g : G) (x : L), (e g) x = g • x) : IsGaloisGroup G K L

instance IsGaloisGroup.fixedPoints (G : Type u_1) (L : Type u_4) [Group G] [Field L] [MulSemiringAction G L] [Finite G] [FaithfulSMul G L] : IsGaloisGroup G (↥(FixedPoints.subfield G L)) L

instance IsGaloisGroup.intermediateField (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) [Finite G] [hGKL : IsGaloisGroup G K L] : IsGaloisGroup (↥(fixingSubgroup G ↑F)) (↥F) L

@[simp]

theorem IsGaloisGroup.card_fixingSubgroup_eq_finrank (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) [Finite G] [IsGaloisGroup G K L] : Nat.card ↥(fixingSubgroup G ↑F) = Module.finrank (↥F) L

theorem IsGaloisGroup.fixingSubgroup_le_of_le (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F F' : IntermediateField K L) (h : F ≤ F') : fixingSubgroup G ↑F' ≤ fixingSubgroup G ↑F

@[simp]

theorem IsGaloisGroup.fixingSubgroup_bot (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [SMulCommClass G K L] : fixingSubgroup G ↑⊥ = ⊤

@[simp]

theorem IsGaloisGroup.fixedPoints_bot (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [SMulCommClass G K L] : FixedPoints.intermediateField ↥⊥ = ⊤

theorem IsGaloisGroup.le_fixedPoints_iff_le_fixingSubgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) (F : IntermediateField K L) [SMulCommClass G K L] : F ≤ FixedPoints.intermediateField ↥H ↔ H ≤ fixingSubgroup G ↑F

theorem IsGaloisGroup.fixedPoints_le_of_le (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H H' : Subgroup G) [SMulCommClass G K L] (h : H ≤ H') : FixedPoints.intermediateField ↥H' ≤ FixedPoints.intermediateField ↥H

theorem IsGaloisGroup.fixingSubgroup_top (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] : fixingSubgroup G ↑⊤ = ⊥

@[simp]

theorem IsGaloisGroup.fixedPoints_top (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] : FixedPoints.intermediateField ↥⊤ = ⊥

noncomputable def IsGaloisGroup.intermediateFieldEquivSubgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] [Finite G] : IntermediateField K L ≃o (Subgroup G)ᵒᵈ

The Galois correspondence from intermediate fields to subgroups.

Equations

• IsGaloisGroup.intermediateFieldEquivSubgroup G K L = IsGalois.intermediateFieldEquivSubgroup.trans (IsGaloisGroup.mulEquivAlgEquiv G K L).comapSubgroup.dual

@[simp]

theorem IsGaloisGroup.intermediateFieldEquivSubgroup_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] [Finite G] {F : IntermediateField K L} : (intermediateFieldEquivSubgroup G K L) F = OrderDual.toDual (fixingSubgroup G ↑F)

theorem IsGaloisGroup.ofDual_intermediateFieldEquivSubgroup_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] [Finite G] {F : IntermediateField K L} : OrderDual.ofDual ((intermediateFieldEquivSubgroup G K L) F) = fixingSubgroup G ↑F

@[simp]

theorem IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] [Finite G] {H : (Subgroup G)ᵒᵈ} : (intermediateFieldEquivSubgroup G K L).symm H = FixedPoints.intermediateField ↥(OrderDual.ofDual H)

theorem IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply_toDual (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] [hGKL : IsGaloisGroup G K L] [Finite G] {H : Subgroup G} : (intermediateFieldEquivSubgroup G K L).symm (OrderDual.toDual H) = FixedPoints.intermediateField ↥H

@[simp]

theorem IsGaloisGroup.fixingSubgroup_fixedPoints (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) [hGKL : IsGaloisGroup G K L] [Finite G] : fixingSubgroup G ↑(FixedPoints.intermediateField ↥H) = H

@[simp]

theorem IsGaloisGroup.fixedPoints_fixingSubgroup (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) [hGKL : IsGaloisGroup G K L] [Finite G] : FixedPoints.intermediateField ↥(fixingSubgroup G ↑F) = F

theorem IsGaloisGroup.fixedPoints_eq_range_algebraMap (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) [hGKL : IsGaloisGroup G K L] [Finite G] (B : Type u_5) [CommSemiring B] [Algebra B L] [IsGaloisGroup (↥H) B L] : ↑(FixedPoints.intermediateField ↥H) = Set.range ⇑(algebraMap B L)

If G acts as a Galois group on L/K and the subgroup H acts as a Galois group on L/B, then the fixed points of H equals the range of algebraMap B L.

theorem IsGaloisGroup.fixingSubgroup_range_algebraMap' (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) [hGKL : IsGaloisGroup G K L] [Finite G] (B : Type u_5) [CommSemiring B] [Algebra B L] [IsGaloisGroup (↥H) B L] : fixingSubgroup G (Set.range ⇑(algebraMap B L)) = H

If G acts as a Galois group on L/K and the subgroup H acts as a Galois group on L/B, then the fixing subgroup of algebraMap B L inside G equals H. See fixingSubgroup_range_algebraMap for a more general version.

theorem IsGaloisGroup.fixingSubgroup_range_algebraMap (G : Type u_1) [Group G] [Finite G] (A : Type u_5) (B : Type u_6) (C : Type u_7) (H : Subgroup G) [CommRing A] [CommRing B] [CommRing C] [IsDomain C] [Algebra A C] [FaithfulSMul A C] [MulSemiringAction G C] [hGAC : IsGaloisGroup G A C] [Algebra B C] [FaithfulSMul B C] [hH : IsGaloisGroup (↥H) B C] : fixingSubgroup G (Set.range ⇑(algebraMap B C)) = H

If G acts on a domain C with IsGaloisGroup G A C, and a subgroup H acts on C with IsGaloisGroup H B C, then the fixing subgroup of algebraMap B C equals H.

@[implicit_reducible]

noncomputable def IsGaloisGroup.smulOfNormal (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [N.Normal] [IsGaloisGroup (↥N) B C] : SMul G B

If N is a normal subgroup of G and IsGaloisGroup N B C, then G acts on B. For g : G and x : B, g • x is the unique element of B whose image in C is g • algebraMap B C x, see algebraMap_smulOfNormal.

Equations

• IsGaloisGroup.smulOfNormal G B C N = { smul := fun (g : G) (x : B) => Exists.choose ⋯ }

@[simp]

theorem IsGaloisGroup.algebraMap_smulOfNormal (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [N.Normal] [IsGaloisGroup (↥N) B C] (g : G) (x : B) : (algebraMap B C) (g • x) = g • (algebraMap B C) x

instance IsGaloisGroup.smulDistribClass_smulOfNormal (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [N.Normal] [IsGaloisGroup (↥N) B C] : SMulDistribClass G B C

If N is normal and IsGaloisGroup N B C, the action smulOfNormal G B C satisfies SMulDistribClass G B C.

@[implicit_reducible]

noncomputable def IsGaloisGroup.mulSemiringActionOfNormal (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [FaithfulSMul B C] [IsGaloisGroup (↥N) B C] [N.Normal] : MulSemiringAction G B

If N is a normal subgroup of G and IsGaloisGroup N B C, then G acts on B as a MulSemiringAction, via the action defined in smulOfNormal.

Equations

• IsGaloisGroup.mulSemiringActionOfNormal G B C N = mulSemiringActionOfSmulDistribClass B C G

@[implicit_reducible]

noncomputable def IsGaloisGroup.mulSemiringActionQuotient (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [FaithfulSMul B C] [IsGaloisGroup (↥N) B C] [N.Normal] : MulSemiringAction (G ⧸ N) B

If N is a normal subgroup of G and IsGaloisGroup N B C, then the quotient group G ⧸ N acts on B by (g : G ⧸ N) • x = g • x.

Equations

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

theorem IsGaloisGroup.mulSemiringActionQuotient_smul_def (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [FaithfulSMul B C] [MulSemiringAction G B] [SMulDistribClass G B C] [IsGaloisGroup (↥N) B C] [N.Normal] (g : G) (b : B) : ↑g • b = g • b

theorem IsGaloisGroup.isScalarTower_mulSemiringActionQuotient (G : Type u_1) [Group G] (B : Type u_6) (C : Type u_7) [Semiring C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [FaithfulSMul B C] [MulSemiringAction G B] [SMulDistribClass G B C] [IsGaloisGroup (↥N) B C] [N.Normal] : IsScalarTower G (G ⧸ N) B

@[implicit_reducible]

def IsGaloisGroup.smulCommClassQuotient (G : Type u_1) [Group G] (A : Type u_5) (B : Type u_6) (C : Type u_7) [CommSemiring A] [Semiring C] [Algebra A C] [MulSemiringAction G C] (N : Subgroup G) [CommSemiring B] [Algebra B C] [FaithfulSMul B C] [N.Normal] [Algebra A B] [IsScalarTower A B C] [SMulCommClass G A C] [MulSemiringAction G B] [MulAction (G ⧸ N) B] [SMulDistribClass G B C] [IsScalarTower G (G ⧸ N) B] : SMulCommClass (G ⧸ N) A B

If G acts on C commuting with A, then the action of G ⧸ N on B commutes with A.

Equations

• ⋯ = ⋯

theorem IsGaloisGroup.quotient (G : Type u_1) [Group G] (A : Type u_5) (B : Type u_6) (C : Type u_7) [CommRing A] [CommRing B] [CommRing C] [IsDomain C] [Algebra A B] [Algebra A C] [Algebra B C] [FaithfulSMul A C] [FaithfulSMul B C] [IsScalarTower A B C] [Finite G] (N : Subgroup G) [N.Normal] [MulSemiringAction G C] [hG : IsGaloisGroup G A C] [MulSemiringAction G B] [MulSemiringAction (G ⧸ N) B] [SMulCommClass (G ⧸ N) A B] [SMulDistribClass G B C] [IsScalarTower G (G ⧸ N) B] [IsGaloisGroup (↥N) B C] : IsGaloisGroup (G ⧸ N) A B

If G is a Galois group for C/A, and the normal subgroup N ≤ G is a Galois group for C/B, then the quotient G ⧸ N is a Galois group for B/A.

@[implicit_reducible]

noncomputable instance IsGaloisGroup.instMulSemiringActionQuotientSubgroupSubtypeMemIntermediateField (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) (N : Subgroup G) [N.Normal] [IsGaloisGroup (↥N) (↥F) L] : MulSemiringAction (G ⧸ N) ↥F

Equations

• IsGaloisGroup.instMulSemiringActionQuotientSubgroupSubtypeMemIntermediateField G K L F N = IsGaloisGroup.mulSemiringActionQuotient G (↥F) L N

instance IsGaloisGroup.instSMulCommClassQuotientSubgroupSubtypeMemIntermediateFieldOfSMulDistribClassOfIsScalarTower (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) (N : Subgroup G) [N.Normal] [IsGaloisGroup (↥N) (↥F) L] [SMulCommClass G K L] [MulSemiringAction G ↥F] [SMulDistribClass G (↥F) L] [IsScalarTower G (G ⧸ N) ↥F] : SMulCommClass (G ⧸ N) K ↥F

instance IsGaloisGroup.instQuotientSubgroupSubtypeMemIntermediateFieldOfFinite (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (F : IntermediateField K L) (N : Subgroup G) [N.Normal] [IsGaloisGroup (↥N) (↥F) L] [Finite G] [IsGaloisGroup G K L] : IsGaloisGroup (G ⧸ N) K ↥F

If G is a finite Galois group for L/K and N is a normal subgroup of G that is a Galois group for L/F, then the quotient group G ⧸ N is a Galois group for F/K.

theorem IsGaloisGroup.map_quotientMk' (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) (F : IntermediateField K L) (N : Subgroup G) [N.Normal] [IsGaloisGroup (↥N) (↥F) L] (E : IntermediateField K L) [hE : IsGaloisGroup (↥H) (↥E) L] [Finite G] [IsGaloisGroup G K L] (h : E ≤ F) : IsGaloisGroup ↥(Subgroup.map (QuotientGroup.mk' N) H) ↥E ↥F

If G is a finite Galois group for L/K, N is a normal subgroup that is a Galois group for L/F, and H is a subgroup that is a Galois group for L/E with E ≤ F, then the image of H under the canonical quotient map G → G ⧸ N is a Galois group for F/E.

@[deprecated IsGaloisGroup.map_quotientMk' (since := "2026-04-21")]

theorem IsGaloisGroup.quotientMap (G : Type u_1) (K : Type u_3) (L : Type u_4) [Group G] [Field K] [Field L] [Algebra K L] [MulSemiringAction G L] (H : Subgroup G) (F : IntermediateField K L) (N : Subgroup G) [N.Normal] [IsGaloisGroup (↥N) (↥F) L] (E : IntermediateField K L) [hE : IsGaloisGroup (↥H) (↥E) L] [Finite G] [IsGaloisGroup G K L] (h : E ≤ F) : IsGaloisGroup ↥(Subgroup.map (QuotientGroup.mk' N) H) ↥E ↥F

Alias of IsGaloisGroup.map_quotientMk'.

---

If G is a finite Galois group for L/K, N is a normal subgroup that is a Galois group for L/F, and H is a subgroup that is a Galois group for L/E with E ≤ F, then the image of H under the canonical quotient map G → G ⧸ N is a Galois group for F/E.