Theorems · Theorem · field theory
Polynomial.Splits.coeff_zero_eq_prod_roots_of_monic
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R],
f.Splits → f.Monic → f.coeff 0 = (-1) ^ f.natDegree * f.roots.prodIf f is a monic polynomial that splits, then coeff f 0 equals the product of the roots.
- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Polynomialstatement and proof · cited by 5,681
- mul_oneproof · cited by 3,885
- IsDomainstatement and proof · cited by 2,196
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffstatement · cited by 1,045
- Multiset.prodstatement and proof · cited by 528
- Polynomial.Monicstatement and proof · cited by 461
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement and proof · cited by 264
- Polynomial.Monic.leadingCoeffproof · cited by 101
- Polynomial.Splits.coeff_zero_eq_leadingCoeff_mul_prod_rootsproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.det_eq_prod_roots_charpoly_of_splitsproof · cited by 3
- Algebra.PowerBasis.norm_gen_eq_prod_rootsproof · cited by 2
- spectralNorm.spectralNorm_eq_norm_coeff_zero_rpowproof · cited by 0