Theorems · Theorem · commutative algebra
minpoly.IsIntegrallyClosed.isIntegral_iff_isUnit_leadingCoeff
∀ {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] {x : S} {p : Polynomial R},
Irreducible p → (Polynomial.aeval x) p = 0 → (IsIntegral R x ↔ IsUnit p.leadingCoeff)If x is a root of an irreducible polynomial p, then x is integral
iff the leading coefficient of p is a unit.
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- Foundations
- Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound
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- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Polynomialstatement and proof · cited by 5,681
- AlgHomstatement · cited by 3,236
- one_mulproof · cited by 2,841
- IsDomainstatement and proof · cited by 2,196
- Units.valproof · cited by 1,966
- IsUnitstatement and proof · cited by 1,602
- one_smulproof · cited by 1,374
- Polynomial.aevalstatement and proof · cited by 615
- Module.IsTorsionFreestatement and proof · cited by 600
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