Theorems · Theorem · field theory
perfectClosure.eq_bot_of_isSeparable
∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.IsSeparable F E], perfectClosure F E = ⊥
If E / F is separable, then the perfect closure of F in E is equal to F. Note that
the converse is not necessarily true (see https://math.stackexchange.com/a/3009197)
even when E / F is algebraic.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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 · cited by 4,720
- IntermediateFieldstatement · cited by 988
- Algebra.IsSeparablestatement and proof · cited by 210
- perfectClosurestatement and proof · cited by 14
- IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparableproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- perfectField_of_isSeparable_of_perfectField_topproof · cited by 1