Theorems · Theorem · field theory
Polynomial.splits_iff_card_roots
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R], f.Splits ↔ f.roots.card = f.natDegreeA polynomial splits if and only if it has as many roots as its degree.
- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- IsDomainstatement and proof · cited by 2,196
- Polynomial.natDegreestatement and proof · cited by 1,105
- Multiset.cardstatement and proof · cited by 375
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement and proof · cited by 264
- Polynomial.Splits.natDegree_eq_card_rootsproof · cited by 16
- Polynomial.splits_iff_exists_multisetproof · cited by 9
- Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_Cproof · cited by 3
Cited by11
Results whose statement or proof uses this declaration.
- Polynomial.norm_coeff_le_choose_mul_mahlerMeasureproof · cited by 3
- X_pow_sub_C_splits_of_isPrimitiveRootproof · cited by 2
- Matrix.IsHermitian.splits_charpolyproof · cited by 2
- LinearMap.IsSymmetric.splits_charpolyproof · cited by 2
- Cubic.splits_iff_card_rootsproof · cited by 2
- Subfield.splits_botproof · cited by 1
- Polynomial.X_pow_sub_one_splitsproof · cited by 1
- spectralNorm.spectralNorm_pow_natDegree_eq_prod_rootsproof · cited by 1
- Polynomial.coeff_eq_esymm_roots_of_splitsproof · cited by 1
- ConjRootClass.minpoly.map_eq_prodproof · cited by 0
- GaloisField.splits_zmod_X_pow_sub_Xproof · cited by 0