Theorems · Theorem · commutative algebra

minpoly.isIntegrallyClosed_dvd

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [IsDomain R] [inst_3 : Algebra R S]
  [IsIntegrallyClosed R] [IsDomain S] [Module.IsTorsionFree R S] {s : S},
  IsIntegral R s → ∀ {p : Polynomial R}, (Polynomial.aeval s) p = 0 → minpoly R s ∣ p

For integrally closed rings, the minimal polynomial divides any polynomial that has the integral element as root. See also minpoly.dvd which relaxes the assumptions on S in exchange for stronger assumptions on R.

Defined in

Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Cited by

9 results in Mathlib

Foundations

Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingIsDomainAlgebraIsIntegrallyClosedIsDomainModule.IsTorsionFree