Theorems · Theorem · commutative algebra

IsIntegrallyClosed.minpoly.unique

∀ {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.Monic →
    (Polynomial.aeval s) P = 0 →
      (∀ (Q : Polynomial R), Q.Monic → (Polynomial.aeval s) Q = 0 → P.degree ≤ Q.degree) → P = minpoly R s

The minimal polynomial of an element x is uniquely characterized by its defining property: if there is another monic polynomial of minimal degree that has x as a root, then this polynomial is equal to the minimal polynomial of x. See also minpoly.unique which relaxes the assumptions on S in exchange for stronger assumptions on R.

Defined in

Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Cited by

2 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingIsDomainAlgebraIsIntegrallyClosedIsDomainModule.IsTorsionFree