Theorems · Theorem · field theory
Polynomial.Splits.eval_eq_prod_roots
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R],
f.Splits → ∀ (x : R), Polynomial.eval x f = f.leadingCoeff * (Multiset.map (fun x_1 => x - x_1) f.roots).prod- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
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
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- Multiset.mapstatement and proof · cited by 876
- Polynomial.evalstatement and proof · cited by 796
- Multiset.prodstatement and proof · cited by 528
- Polynomial.leadingCoeffstatement and proof · cited by 498
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement and proof · cited by 264
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Splits.eval_eq_prod_roots_of_monicproof · cited by 2
- Polynomial.Splits.eval_derivative_eq_eval_mul_sumproof · cited by 1
- Polynomial.Splits.aeval_eq_prod_arootsproof · cited by 0