Theorems · Theorem · field theory
Polynomial.Splits.nextCoeff_eq_neg_sum_roots_of_monic
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R],
f.Splits → f.Monic → f.nextCoeff = -f.roots.sumIf f is a monic polynomial that splits, then f.nextCoeff equals the negative of the sum
of the roots.
- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 2 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.
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
- one_mulproof · cited by 2,841
- IsDomainstatement and proof · cited by 2,196
- neg_mulproof · cited by 654
- Polynomial.Monicstatement and proof · cited by 461
- Multiset.sumstatement and proof · cited by 388
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement and proof · cited by 264
- Polynomial.Monic.leadingCoeffproof · cited by 101
- Polynomial.nextCoeffstatement · cited by 46
- Polynomial.Splits.nextCoeff_eq_neg_sum_roots_mul_leadingCoeffproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.trace_eq_sum_roots_charpoly_of_splitsproof · cited by 3
- PowerBasis.trace_gen_eq_sum_rootsproof · cited by 2