Theorems · Theorem · field theory
separableClosure.eq_bot_iff
∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.IsAlgebraic F E],
separableClosure F E = ⊥ ↔ IsPurelyInseparable F EIf E / F is an algebraic extension, then the separable closure of F in E is
equal to F if and only if E / F is purely inseparable.
- Cited by
- 1 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.
Cites12
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
- Fieldstatement and proof · cited by 7,404
- Bot.botstatement and proof · cited by 4,720
- IntermediateFieldstatement · cited by 988
- Algebra.IsAlgebraicstatement and proof · cited by 322
- Algebra.IsIntegral.isIntegralproof · cited by 86
- IsPurelyInseparablestatement and proof · cited by 84
- IsSeparableproof · cited by 68
- separableClosurestatement and proof · cited by 55
- mem_separableClosure_iffproof · cited by 7
- isPurelyInseparable_iffproof · cited by 4
- separableClosure.eq_bot_of_isPurelyInseparableproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- isSepClosed_iff_isPurelyInseparable_algebraicClosureproof · cited by 1