Theorems · Theorem · field theory

IsGaloisGroup.quotient

∀ (G : Type u_1) [inst : Group G] (A : Type u_5) (B : Type u_6) (C : Type u_7) [inst_1 : CommRing A]
  [inst_2 : CommRing B] [inst_3 : CommRing C] [IsDomain C] [inst_5 : Algebra A B] [inst_6 : Algebra A C]
  [inst_7 : Algebra B C] [FaithfulSMul A B] [FaithfulSMul B C] [IsScalarTower A B C] [Finite G] (N : Subgroup G)
  [inst_12 : N.Normal] [inst_13 : MulSemiringAction G C] [hG : IsGaloisGroup G A C] [inst_14 : MulSemiringAction G B]
  [inst_15 : 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.

Defined in

Mathlib.FieldTheory.Galois.IsGaloisGroup

Cited by

1 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupCommRingCommRingCommRingIsDomainAlgebraAlgebraAlgebraFaithfulSMulFaithfulSMulIsScalarTowerFiniteSubgroup.NormalMulSemiringActionIsGaloisGroupMulSemiringActionMulSemiringActionSMulCommClassSMulDistribClassIsScalarTowerIsGaloisGroup