Theorems · Theorem · field theory
Polynomial.Splits.natDegree_eq_card_roots
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R} [inst_1 : IsDomain R], f.Splits → f.natDegree = f.roots.card- Defined in
- Mathlib.Algebra.Polynomial.Splits
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Polynomial.leadingCoeffproof · cited by 498
- Multiset.cardstatement and proof · cited by 375
- Polynomial.Splitsstatement and proof · cited by 290
- Polynomial.rootsstatement and proof · cited by 264
- Polynomial.leadingCoeff_eq_zeroproof · cited by 55
- Polynomial.roots.congr_simpproof · cited by 31
- Polynomial.roots_zeroproof · cited by 19
- Polynomial.Splits.eq_prod_rootsproof · cited by 16
Cited by16
Results whose statement or proof uses this declaration.
- Polynomial.splits_iff_card_rootsproof · cited by 11
- Polynomial.resultant_eq_prod_evalproof · cited by 3
- Polynomial.card_rootSet_eq_natDegreeproof · cited by 3
- AlgHom.natCard_of_powerBasisproof · cited by 2
- Polynomial.eq_one_of_roots_leproof · cited by 2
- Polynomial.Splits.roots_map_of_injectiveproof · cited by 2
- Polynomial.coeff_le_of_roots_leproof · cited by 1
- Polynomial.resultant_eq_prod_roots_subproof · cited by 1
- Polynomial.Gal.galActionHom_bijective_of_prime_degreeproof · cited by 1
- Polynomial.resultant_scaleRootsproof · 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