Theorems · Theorem · field theory
Polynomial.leadingCoeff_ne_zero
∀ {R : Type u} [inst : Semiring R] {p : Polynomial R}, p.leadingCoeff ≠ 0 ↔ p ≠ 0- Defined in
- Mathlib.Algebra.Polynomial.Degree.Defs
- Cited by
- 39 results in Mathlib
- Foundations
- Depth 90 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.
Cites4
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.leadingCoeffstatement and proof · cited by 498
- Polynomial.leadingCoeff_eq_zeroproof · cited by 55
Cited by39
Results whose statement or proof uses this declaration.
- RatFunc.num_div_denomproof · cited by 18
- Polynomial.degree_map_eq_of_leadingCoeff_ne_zeroproof · cited by 9
- minpoly.eq_of_irreducibleproof · cited by 8
- Polynomial.irreducible_mul_leadingCoeff_invproof · cited by 6
- Polynomial.nextCoeff_eq_zero_of_eraseLead_eq_zeroproof · cited by 5
- Polynomial.associated_of_dvd_of_natDegree_leproof · cited by 4
- Polynomial.coeffList_eraseLeadproof · cited by 4
- Polynomial.induction_with_natDegree_leproof · cited by 4
- Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_rootsproof · cited by 4
- RatFunc.eq_C_of_minpolyX_coeff_eq_zeroproof · cited by 3
- Polynomial.natDegree_of_dvd_cyclotomic_of_irreducibleproof · cited by 3
- Polynomial.exists_monic_irreducible_factorproof · cited by 3