Theorems · Theorem · commutative algebra

IsGaloisGroup.iff_isFractionRing

∀ (G : Type u_1) (A : Type u_2) (B : Type u_3) (K : Type u_4) (L : Type u_5) [inst : Group G] [inst_1 : CommRing A]
  [inst_2 : CommRing B] [inst_3 : MulSemiringAction G B] [inst_4 : Algebra A B] [inst_5 : Field K] [inst_6 : Field L]
  [inst_7 : Algebra K L] [inst_8 : Algebra A K] [inst_9 : Algebra B L] [inst_10 : Algebra A L] [IsFractionRing A K]
  [IsFractionRing B L] [IsScalarTower A K L] [IsScalarTower A B L] [inst_15 : 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.

Defined in

Mathlib.RingTheory.IsGaloisGroup.Basic

Cited by

0 results in Mathlib

Foundations

Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupCommRingCommRingMulSemiringActionAlgebraFieldFieldAlgebraAlgebraAlgebraAlgebraIsFractionRingIsFractionRingIsScalarTowerIsScalarTowerMulSemiringActionSMulDistribClassFiniteIsIntegrallyClosed