Theorems · Theorem · commutative algebra
minpoly.IsIntegrallyClosed.isIntegral_iff_leadingCoeff_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} {p : Polynomial R},
(Polynomial.aeval s) p = 0 →
p ≠ 0 →
(∀ (q : Polynomial R), q.Monic → (Polynomial.aeval s) q = 0 → p.degree ≤ q.degree) →
(IsIntegral R s ↔ Polynomial.C p.leadingCoeff ∣ p)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound
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- Polynomialstatement and proof · cited by 5,681
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- IsDomainstatement and proof · cited by 2,196
- Polynomial.Cstatement and proof · cited by 1,598
- WithBotstatement and proof · cited by 1,498
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