Theorems · Theorem · commutative algebra
isIntegral_leadingCoeff_smul
∀ {R : Type u_1} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] (p : Polynomial R) (x : S)
[inst_2 : Algebra R S], (Polynomial.aeval x) p = 0 → IsIntegral R (p.leadingCoeff • x)Given a p : R[X] and a root x : S,
then p.leadingCoeff • x : S is integral over R.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Algebra.algebraMapproof · cited by 4,706
- AlgHomstatement · cited by 3,236
- Polynomial.aevalstatement and proof · cited by 615
- Polynomial.leadingCoeffstatement and proof · cited by 498
- IsIntegralstatement and proof · cited by 427
- Algebra.smul_defproof · cited by 287
- Polynomial.aeval_defproof · cited by 51
- RingHom.isIntegralElem_leadingCoeff_mulproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- IsLocalization.Away.exists_isIntegral_mul_of_isIntegral_algebraMapproof · cited by 2
- minpoly.IsIntegrallyClosed.isIntegral_iff_isUnit_leadingCoeffproof · cited by 0