Theorems · Theorem · commutative algebra
IsIntegral.of_aeval_monic_of_isIntegral_coeff
∀ {R : Type u_4} {A : Type u_5} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {x : A}
{p : Polynomial A},
p.Monic →
p.natDegree ≠ 0 → IsIntegral R (Polynomial.eval x p) → (∀ (i : ℕ), IsIntegral R (p.coeff i)) → IsIntegral R x- Defined in
- Mathlib.RingTheory.Polynomial.IsIntegral
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffstatement and proof · cited by 1,045
- Polynomial.mapproof · cited by 806
- Polynomial.evalstatement and proof · cited by 796
- Polynomial.Monicstatement and proof · cited by 461
- IsIntegralstatement and proof · cited by 427
- integralClosureproof · cited by 105
- IsIntegral.tower_topproof · cited by 30
Cited by1
Results whose statement or proof uses this declaration.
- exists_derivative_mul_eq_and_isIntegral_coeffproof · cited by 1