Theorems · Theorem · field theory
IsSepClosed.separableClosure_eq_bot_iff
∀ (F : Type u_1) (E : Type u_2) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [IsSepClosed E], separableClosure F E = ⊥ ↔ IsSepClosed F
If E / F is a field extension and E is separably closed, then the separable closure
of F in E is equal to F if and only if F is separably closed.
- Defined in
- Mathlib.FieldTheory.IsSepClosed
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldFieldAlgebraIsSepClosed
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- Polynomialproof · cited by 5,681
- Bot.botstatement and proof · cited by 4,720
- LT.lt.ne'proof · cited by 1,417
- IntermediateFieldstatement · cited by 988
- Polynomial.aevalproof · cited by 615
- Irreducibleproof · cited by 496
- AlgHom.toRingHomproof · cited by 490
- Polynomial.Monicproof · cited by 461
- Algebra.ofIdproof · cited by 166
Cited by1
Results whose statement or proof uses this declaration.
- isSepClosed_iff_isPurelyInseparable_algebraicClosureproof · cited by 1