Theorems · Theorem · commutative algebra

Algebra.discr_mul_isIntegral_mem_adjoin

∀ (K : Type u) {L : Type v} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] [Module.Finite K L] {R : Type z}
  [inst_4 : CommRing R] [inst_5 : Algebra R K] [inst_6 : Algebra R L] [IsScalarTower R K L] [Algebra.IsSeparable K L]
  [IsIntegrallyClosed R] [IsFractionRing R K] {B : PowerBasis K L},
  IsIntegral R B.gen → ∀ {z : L}, IsIntegral R z → Algebra.discr K ⇑B.basis • z ∈ R[B.gen]

Let K be the fraction field of an integrally closed domain R and let L be a finite separable extension of K. Let B : PowerBasis K L be such that IsIntegral R B.gen. Then for all, z : L that are integral over R, we have (discr K B.basis) • z ∈ adjoin R ({B.gen} : Set L).

Defined in

Mathlib.RingTheory.Discriminant

Cited by

1 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraModule.FiniteCommRingAlgebraAlgebraIsScalarTowerAlgebra.IsSeparableIsIntegrallyClosedIsFractionRing