Theorems · Theorem · field theory
Polynomial.natSepDegree_eq_of_isAlgClosed
∀ {F : Type u} (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (f : Polynomial F)
[inst_3 : DecidableEq E] [IsAlgClosed E], f.natSepDegree = (f.aroots E).toFinset.cardThe separable degree of a polynomial is equal to the number of distinct roots of it over any algebraically closed field.
- Defined in
- Mathlib.FieldTheory.SeparableDegree
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Algebra.algebraMapproof · cited by 4,706
- Finset.cardstatement · cited by 2,327
- Polynomial.mapproof · cited by 806
- Multiset.toFinsetstatement · cited by 230
- IsAlgClosedstatement and proof · cited by 150
- Polynomial.arootsstatement · cited by 89
- Polynomial.natSepDegreestatement · cited by 53
- IsAlgClosed.splitsproof · cited by 29
- Polynomial.natSepDegree_eq_of_splitsproof · cited by 2
Cited by9
Results whose statement or proof uses this declaration.
- Polynomial.natSepDegree_expandproof · cited by 4
- Polynomial.natSepDegree_powproof · cited by 4
- Polynomial.natSepDegree_le_of_dvdproof · cited by 3
- Polynomial.natSepDegree_mapproof · cited by 1
- Polynomial.natSepDegree_mul_eq_iffproof · cited by 1
- IntermediateField.finSepDegree_adjoin_simple_eq_natSepDegreeproof · cited by 1
- Polynomial.natSepDegree_C_mulproof · cited by 1
- Polynomial.natSepDegree_mulproof · cited by 0
- Polynomial.natSepDegree_smul_nonzeroproof · cited by 0