Theorems · Theorem · field theory
IntermediateField.eq_bot_of_isSepClosed_of_isSeparable
∀ {k : Type u} [inst : Field k] {K : Type v} [inst_1 : Field K] [IsSepClosed k] [inst_3 : Algebra k K]
(L : IntermediateField k K) [Algebra.IsSeparable k ↥L], L = ⊥If k is separably closed, K / k is a field extension, L / k is an intermediate field
which is separable, then L is equal to k. A corollary of IsSepClosed.algebraMap_surjective.
- Defined in
- Mathlib.FieldTheory.IsSepClosed
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Bot.botstatement · cited by 4,720
- Algebra.algebraMapproof · cited by 4,706
- IntermediateFieldstatement and proof · cited by 988
- Algebra.IsSeparablestatement and proof · cited by 210
- bot_uniqueproof · cited by 57
- IsSepClosedstatement and proof · cited by 41
- IsSepClosed.algebraMap_surjectiveproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- IsSepClosed.separableClosure_eq_bot_iffproof · cited by 1