Theorems · Theorem · field theory
Polynomial.natSepDegree_eq_of_splits
∀ {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], (Polynomial.map (algebraMap F E) f).Splits → f.natSepDegree = (f.aroots E).toFinset.cardIf a polynomial splits over E, then its separable degree is equal to
the number of distinct roots of it over E.
- Defined in
- Mathlib.FieldTheory.SeparableDegree
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldFieldAlgebraDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Finsetproof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- RingHomproof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Algebra.algebraMapstatement and proof · cited by 4,706
- Multisetproof · cited by 2,627
- Finset.cardstatement and proof · cited by 2,327
- Polynomial.mapstatement and proof · cited by 806
- RingHomClass.toRingHomproof · cited by 746
- Polynomial.Splitsstatement and proof · cited by 290
Cited by2
Results whose statement or proof uses this declaration.
- Polynomial.natSepDegree_eq_of_isAlgClosedproof · cited by 9
- Polynomial.Monic.eq_X_sub_C_pow_of_natSepDegree_eq_one_of_splitsproof · cited by 0