Theorems · Theorem · commutative algebra

mem_adjoin_of_dvd_coeff_of_dvd_aeval

∀ {A : Type u_1} {B : Type u_2} [inst : CommSemiring A] [inst_1 : Ring B] [inst_2 : Algebra A B] [IsDomain A]
  [Module.IsTorsionFree A B] {Q : Polynomial A} {p : A} {x z : B},
  p ≠ 0 → (∀ i ∈ Finset.range (Q.natDegree + 1), p ∣ Q.coeff i) → (Polynomial.aeval x) Q = p • z → z ∈ A[x]

Defined in

Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Cited by

1 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringRingAlgebraIsDomainModule.IsTorsionFree