Theorems · Theorem · field theory
Field.finSepDegree_eq
∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.IsAlgebraic F E], Field.finSepDegree F E = Cardinal.toNat (Field.sepDegree F E)
If E / F is algebraic, then the Field.finSepDegree F E is equal to Field.sepDegree F E
as a natural number. This means that the cardinality of Field.Emb F E and the degree of
(separableClosure F E) / F are both finite or infinite, and when they are finite, they
coincide.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- mul_oneproof · cited by 3,885
- Cardinalstatement · cited by 2,598
- Module.finrankproof · cited by 1,770
- MonoidWithZeroHomstatement · cited by 704
- Algebra.IsAlgebraicstatement and proof · cited by 322
- Cardinal.toNatstatement · cited by 153
- separableClosureproof · cited by 55
- Field.sepDegreestatement · cited by 24
- Field.finSepDegreestatement and proof · cited by 24
Cited by2
Results whose statement or proof uses this declaration.
- minpoly.map_eq_of_isSeparable_of_isPurelyInseparableproof · cited by 1
- Field.finSepDegree_mul_finInsepDegreeproof · cited by 0