Theorems · Theorem · field theory
Polynomial.Splits.eq_prod_roots
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R],
f.Splits → f = Polynomial.C f.leadingCoeff * (Multiset.map (fun x => Polynomial.X - Polynomial.C x) f.roots).prod- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 137 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- mul_oneproof · cited by 3,885
- Multisetproof · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xstatement and proof · cited by 1,639
- map_zeroproof · cited by 1,614
- Polynomial.Cstatement and proof · cited by 1,598
- Multiset.mapstatement and proof · cited by 876
- Multiset.prodstatement and proof · cited by 528
Cited by16
Results whose statement or proof uses this declaration.
- Polynomial.Splits.natDegree_eq_card_rootsproof · cited by 16
- Polynomial.Splits.eq_prod_roots_of_monicproof · cited by 4
- Polynomial.Splits.eval_eq_prod_rootsproof · cited by 3
- Polynomial.map_sub_sprod_roots_eq_prod_map_evalproof · cited by 3
- spectrum.map_polynomial_aeval_of_degree_posproof · cited by 2
- Polynomial.splits_X_sub_C_mul_iffproof · cited by 1
- Polynomial.Splits.nextCoeff_eq_neg_sum_roots_mul_leadingCoeffproof · cited by 1
- Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_rootsproof · cited by 1
- Polynomial.Splits.of_splits_map_of_injectiveproof · cited by 1
- Polynomial.Splits.coeff_zero_eq_leadingCoeff_mul_prod_rootsproof · cited by 1
- splits_X_pow_sub_one_of_X_pow_sub_Cproof · cited by 1
- AlgebraicClosure.toSplittingField_coeffproof · cited by 1