Theorems · Definition · commutative algebra
Polynomial.integralNormalization
{R : Type u} → [inst : Semiring R] → Polynomial R → Polynomial RIf p : R[X] is a nonzero polynomial with root z, integralNormalization p is
a monic polynomial with root leadingCoeff f * z.
Moreover, integralNormalization 0 = 0.
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Polynomialstatement and proof · cited by 5,681
- Polynomial.natDegreeproof · cited by 1,105
- Polynomial.degreeproof · cited by 643
- Polynomial.leadingCoeffproof · cited by 498
- Polynomial.monomialproof · cited by 256
- Polynomial.sumproof · cited by 67
Cited by24
Results whose statement or proof uses this declaration.
- IsAlgebraic.exists_integral_multipleproof · cited by 8
- Polynomial.integralNormalization_coeffstatement · cited by 6
- Polynomial.integralNormalization_zerostatement · cited by 5
- Polynomial.support_integralNormalization_subsetstatement · cited by 3
- Polynomial.monic_integralNormalizationstatement and proof · cited by 2
- RingHom.isIntegralElem_leadingCoeff_mulproof · cited by 2
- Polynomial.integralNormalization_coeff_natDegreestatement · cited by 2
- Polynomial.integralNormalization_eval₂_eq_zero_of_commutestatement and proof · cited by 2
- Polynomial.integralNormalization_eval₂_leadingCoeff_mul_of_commutestatement and proof · cited by 2
- Polynomial.integralNormalization_mul_C_leadingCoeffstatement and proof · cited by 2
- Polynomial.degree_integralNormalizationstatement and proof · cited by 2
- Polynomial.integralNormalization_aeval_eq_zerostatement · cited by 1