Theorems · Theorem · field theory
separableClosure.adjoin_eq_of_isAlgebraic
∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (K : Type w) [inst_3 : Field K]
[inst_4 : Algebra F K] [inst_5 : Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E],
IntermediateField.adjoin E ↑(separableClosure F K) = separableClosure E KIf K / E / F is a field extension tower, such that E / F is algebraic, then
E adjoin separableClosure F K is equal to separableClosure E K.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Top.topproof · cited by 9,680
- SetLike.coestatement and proof · cited by 8,199
- Fieldstatement and proof · cited by 7,404
- Set.imageproof · cited by 5,609
- IsScalarTowerstatement and proof · cited by 3,896
- IntermediateFieldstatement and proof · cited by 988
- Set.image_congrproof · cited by 533
- IntermediateField.adjoinstatement and proof · cited by 382
- Algebra.IsAlgebraicstatement and proof · cited by 322
- separableClosurestatement and proof · cited by 55
- IsScalarTower.of_algebraMap_eqproof · cited by 40
Cited by1
Results whose statement or proof uses this declaration.
- Field.lift_rank_mul_lift_insepDegree_of_isPurelyInseparableproof · cited by 2