Theorems · Theorem · commutative algebra

IsFractionRing.isAlgebraic_iff

∀ (A : Type u_3) (K : Type u_4) (C : Type u_5) [inst : CommRing A] [IsDomain A] [inst_2 : Field K]
  [inst_3 : Algebra A K] [IsFractionRing A K] [inst_5 : CommRing C] [inst_6 : Algebra A C] [inst_7 : Algebra K C]
  [IsScalarTower A K C] {x : C}, IsAlgebraic A x ↔ IsAlgebraic K x

An element of a ring is algebraic over the ring A iff it is algebraic over the field of fractions of A.

Defined in

Mathlib.RingTheory.Localization.Integral

Cited by

3 results in Mathlib

Foundations

Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainFieldAlgebraIsFractionRingCommRingAlgebraAlgebraIsScalarTower