Theorems · Theorem · commutative algebra

IsIntegralClosure.isFractionRing_of_finite_extension

∀ (A : Type u_3) (K : Type u_4) [inst : CommRing A] (L : Type u_5) [inst_1 : Field K] [inst_2 : Field L]
  [inst_3 : Algebra A K] [inst_4 : Algebra A L] [IsFractionRing A K] (C : Type u_6) [inst_6 : CommRing C] [IsDomain C]
  [inst_8 : Algebra C L] [IsIntegralClosure C A L] [inst_10 : Algebra A C] [IsScalarTower A C L] [IsDomain A]
  [inst_13 : Algebra K L] [IsScalarTower A K L] [FiniteDimensional K L], IsFractionRing C L

If the field L is a finite extension of the fraction field of the integral domain A, the integral closure C of A in L has fraction field L.

Defined in

Mathlib.RingTheory.Localization.Integral

Cited by

8 results in Mathlib

Foundations

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

Assumes

CommRingFieldFieldAlgebraAlgebraIsFractionRingCommRingIsDomainAlgebraIsIntegralClosureAlgebraIsScalarTowerIsDomainAlgebraIsScalarTowerFiniteDimensional