Theorems · Theorem · field theory
eq_separableClosure_iff
∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.IsAlgebraic F E] (L : IntermediateField F E), L = separableClosure F E ↔ Algebra.IsSeparable F ↥L ∧ IsPurelyInseparable (↥L) E
If E / F is algebraic, then an intermediate field of E / F is equal to the separable closure
of F in E if and only if it is separable over F, and E is purely inseparable
over it.
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites8
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
- IntermediateFieldstatement and proof · cited by 988
- Algebra.IsAlgebraicstatement and proof · cited by 322
- Algebra.IsSeparablestatement and proof · cited by 210
- IsPurelyInseparablestatement and proof · cited by 84
- separableClosurestatement and proof · cited by 55
- eq_separableClosureproof · cited by 1
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