Theorems · Theorem · field theory
Polynomial.roots_mul
∀ {R : Type u} [inst : CommRing R] [inst_1 : IsDomain R] {p q : Polynomial R},
p * q ≠ 0 → (p * q).roots = p.roots + q.roots- Defined in
- Mathlib.Algebra.Polynomial.Roots
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Multisetstatement · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Multiset.countproof · cited by 302
- Polynomial.rootsstatement and proof · cited by 264
- Polynomial.rootMultiplicityproof · cited by 79
- Multiset.count_addproof · cited by 23
- Polynomial.count_rootsproof · cited by 19
- Multiset.extproof · cited by 6
- Polynomial.rootMultiplicity_mulproof · cited by 3
Cited by12
Results whose statement or proof uses this declaration.
- Polynomial.roots_C_mulproof · cited by 9
- Polynomial.roots_powproof · cited by 4
- Polynomial.aroots_mulproof · cited by 4
- Polynomial.roots.le_of_dvdproof · cited by 4
- Polynomial.roots_quadratic_eq_pair_iff_of_ne_zeroproof · cited by 2
- Polynomial.Splits.roots_map_of_ne_zeroproof · cited by 1
- Polynomial.roots_list_prodproof · cited by 1
- Polynomial.exists_prod_multiset_X_sub_C_mulproof · cited by 1
- Polynomial.splits_X_sub_C_mul_iffproof · cited by 1
- Polynomial.roots_countP_pos_le_signVariationsproof · cited by 0
- Polynomial.SplittingFieldAux.adjoin_rootSetproof · cited by 0
- Polynomial.nthRoots_two_oneproof · cited by 0