Theorems · Theorem · field theory
Polynomial.natDegree_le_of_dvd
∀ {R : Type u} [inst : Semiring R] [NoZeroDivisors R] {p q : Polynomial R}, p ∣ q → q ≠ 0 → p.natDegree ≤ q.natDegree- Defined in
- Mathlib.Algebra.Polynomial.Degree.Domain
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Polynomialstatement and proof · cited by 5,681
- Polynomial.natDegreestatement and proof · cited by 1,105
- NoZeroDivisorsstatement and proof · cited by 545
- mul_ne_zero_iffproof · cited by 39
- Polynomial.natDegree_mulproof · cited by 23
Cited by11
Results whose statement or proof uses this declaration.
- AdjoinRoot.minpoly_rootproof · cited by 3
- X_pow_sub_C_irreducible_of_primeproof · cited by 2
- Polynomial.separable_iff_derivative_ne_zeroproof · cited by 2
- Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeffproof · cited by 1
- minpoly.IsIntegrallyClosed.degree_le_of_ne_zeroproof · cited by 1
- IntermediateField.adjoin_minpoly_coeff_of_exists_primitive_elementproof · cited by 1
- Polynomial.Gal.prime_degree_dvd_cardproof · cited by 1
- Polynomial.eq_zero_of_dvd_of_natDegree_ltproof · cited by 1
- Polynomial.degree_gcd_le_leftproof · cited by 1
- minpoly.map_algebraMapproof · cited by 1
- natDegree_radical_leproof · cited by 0