Theorems · Theorem · field theory
Polynomial.Splits.coeff_zero_eq_leadingCoeff_mul_prod_roots
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R],
f.Splits → f.coeff 0 = (-1) ^ f.natDegree * f.leadingCoeff * f.roots.prod- Defined in
- Mathlib.Algebra.Polynomial.Splits
- 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.
Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Polynomialstatement and proof · cited by 5,681
- IsDomainstatement and proof · cited by 2,196
- mul_assocproof · cited by 1,667
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffstatement and proof · cited by 1,045
- Multiset.mapproof · cited by 876
- Polynomial.evalproof · cited by 796
- Multiset.prodstatement and proof · cited by 528
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.Splits.coeff_zero_eq_prod_roots_of_monicproof · cited by 3