Theorems · Theorem · commutative algebra

minpoly.IsIntegrallyClosed.degree_le_of_ne_zero

∀ {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} {p : Polynomial R},
  p ≠ 0 → (Polynomial.aeval s) p = 0 → (minpoly R s).degree ≤ p.degree

If an element x is a root of a nonzero polynomial p, then the degree of p is at least the degree of the minimal polynomial of x. See also minpoly.degree_le_of_ne_zero which relaxes the assumptions on S in exchange for stronger assumptions on R.

Defined in

Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Cited by

1 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingIsDomainAlgebraIsIntegrallyClosedIsDomainModule.IsTorsionFree