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Milneより、下記も関連だな

Separating transcendence bases
THEOREM 9.27 If F is perfect, then every finitely generated extension E of F admits a separating transcendence basis over F .

Transcendental Galois theory
THEOREM 9.29 Let Ω be an algebraically closed field and let F be a perfect subfield of Ω
.
If α ∈ Ω
is fixed by all F -automorphisms of Ω
, then α ∈ F, i.e., Ω^(Aut(Ω/F)
=F.