Theorems · Theorem · field theory
Polynomial.splits_iff_exists_multiset
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R},
f.Splits ↔ ∃ m, f = Polynomial.C f.leadingCoeff * (Multiset.map (fun x => Polynomial.X - Polynomial.C x) m).prod- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Polynomialstatement and proof · cited by 5,681
- Multisetstatement and proof · cited by 2,627
- Polynomial.Xstatement and proof · cited by 1,639
- Polynomial.Cstatement and proof · cited by 1,598
- Multiset.mapstatement and proof · cited by 876
- Multiset.prodstatement and proof · cited by 528
- Polynomial.leadingCoeffstatement and proof · cited by 498
- map_negproof · cited by 378
- Polynomial.Splitsstatement · cited by 290
Cited by9
Results whose statement or proof uses this declaration.
- Polynomial.Splits.eq_prod_rootsproof · cited by 16
- Polynomial.Splits.exists_eval_eq_zeroproof · cited by 14
- Polynomial.splits_iff_card_rootsproof · cited by 11
- spectralNorm_eq_iSup_of_finiteDimensional_normalproof · cited by 2
- Polynomial.isIntegral_coeff_of_factorsproof · cited by 1
- Polynomial.Splits.of_splits_map_of_injectiveproof · cited by 1
- splits_X_pow_sub_one_of_X_pow_sub_Cproof · cited by 1
- exists_derivative_mul_eq_and_isIntegral_coeffproof · cited by 1
- Polynomial.exists_finset_of_splitsproof · cited by 0