le_restrictScalars — Mathlib

English

In a tower F ⊆ E ⊆ K, the separable closure of F in K is contained in the restriction of scalars of the separable closure over E.

Русский

В тюрьме F ⊆ E ⊆ K сепарабельное замыкание F в K содержится в ограничении скаляров сепарабельного замыкания над E.

LaTeX

$$$\\mathrm{separableClosure}(F,K) \\le (\\mathrm{separableClosure}(E,K)).\\mathrm{restrictScalars}(F)$$$

Lean4

/-- If `K / E / F` is a field extension tower, then `separableClosure F K` is contained in
`separableClosure E K`. -/
theorem le_restrictScalars [Algebra E K] [IsScalarTower F E K] :
    separableClosure F K ≤ (separableClosure E K).restrictScalars F := fun _ ↦ IsSeparable.tower_top E

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