Theorems · Definition · field theory

IsGaloisGroup.quotientMulEquiv

(G : Type u_1) →
  (G' : Type u_2) →
    [inst : Group G] →
      [inst_1 : Group G'] →
        (A : Type u_5) →
          (B : Type u_6) →
            (C : Type u_7) →
              [inst_2 : CommRing A] →
                [inst_3 : CommRing B] →
                  [inst_4 : CommRing C] →
                    [IsDomain C] →
                      [inst_6 : Algebra A B] →
                        [inst_7 : Algebra A C] →
                          [inst_8 : Algebra B C] →
                            [FaithfulSMul A B] →
                              [FaithfulSMul B C] →
                                [IsScalarTower A B C] →
                                  [Finite G] →
                                    [Finite G'] →
                                      (N : Subgroup G) →
                                        [inst_14 : N.Normal] →
                                          [inst_15 : MulSemiringAction G C] →
                                            [IsGaloisGroup G A C] →
                                              [IsGaloisGroup (↥N) B C] →
                                                [inst_18 : MulSemiringAction G' B] →
                                                  [IsGaloisGroup G' A B] → G ⧸ N ≃* G'

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

Defined in

Mathlib.FieldTheory.Galois.IsGaloisGroup

Cited by

4 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupGroupCommRingCommRingCommRingIsDomainAlgebraAlgebraAlgebraFaithfulSMulFaithfulSMulIsScalarTowerFiniteFiniteSubgroup.NormalMulSemiringActionIsGaloisGroupIsGaloisGroupMulSemiringActionIsGaloisGroup