Theorems · Theorem · field theory

IntermediateField.exists_finset_maximalFor_isTranscendenceBasis_separableClosure

∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.EssFiniteType F E],
  ∃ s,
    MaximalFor (fun t => IsTranscendenceBasis F Subtype.val)
      (fun t => IntermediateField.restrictScalars F (separableClosure (↥(IntermediateField.adjoin F t)) E)) ↑s

In a finitely generated field extension, there exists a maximal separably generated field extension.

Defined in: Mathlib.FieldTheory.SeparableClosure
Cited by: 1 results in Mathlib
Foundations: Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field Field Algebra Algebra.EssFiniteType

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Cites40

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
Finsetstatement and proof · cited by 13,712
Algebrastatement and proof · cited by 11,388
Top.topproof · cited by 9,680
SetLike.coestatement and proof · cited by 8,199
Fieldstatement and proof · cited by 7,404
Set.ofPredproof · cited by 6,101
Set.Nonemptyproof · cited by 2,627
Set.Finiteproof · cited by 1,814
Module.Finiteproof · cited by 1,032
IntermediateFieldstatement and proof · cited by 988
Algebra.adjoinproof · cited by 535

Cited by1

Results whose statement or proof uses this declaration.

exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow_of_essFiniteTypeproof · cited by 1