Theorems · Theorem · commutative algebra

IsIntegralClosure.isFractionRing_of_algebraic

∀ (A : Type u_3) [inst : CommRing A] {L : Type u_5} [inst_1 : Field L] [inst_2 : Algebra A L] (C : Type u_6)
  [inst_3 : CommRing C] [IsDomain C] [inst_5 : Algebra C L] [IsIntegralClosure C A L] [inst_7 : Algebra A C]
  [IsScalarTower A C L] [Algebra.IsAlgebraic A L], (∀ (x : A), (algebraMap A L) x = 0 → x = 0) → IsFractionRing C L

If the field L is an algebraic extension 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

2 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraCommRingIsDomainAlgebraIsIntegralClosureAlgebraIsScalarTowerAlgebra.IsAlgebraic