Theorems · Theorem · field theory
Polynomial.prod_multiset_X_sub_C_dvd
∀ {R : Type u} [inst : CommRing R] [inst_1 : IsDomain R] (p : Polynomial R),
(Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod ∣ pThe product ∏ (X - a) for a inside the multiset p.roots divides p.
- Defined in
- Mathlib.Algebra.Polynomial.Roots
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites32
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Polynomialstatement and proof · cited by 5,681
- Algebra.algebraMapproof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xstatement and proof · cited by 1,639
- Polynomial.Cstatement and proof · cited by 1,598
- Multiset.mapstatement · cited by 876
- Polynomial.mapproof · cited by 806
- Function.onFunproof · cited by 570
Cited by5
Results whose statement or proof uses this declaration.
- Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_Cproof · cited by 3
- Polynomial.count_map_rootsproof · cited by 2
- Polynomial.exists_prod_multiset_X_sub_C_mulproof · cited by 1
- Polynomial.Splits.dvd_of_roots_le_rootsproof · cited by 1
- Multiset.prod_X_sub_C_dvd_iff_le_rootsproof · cited by 0