Section Automorphism.
End Automorphism.
Definition fixedField V (A : {set gal_of V}) :=
(V :&: \bigcap_(x in A) fixedSpace x)%VS.

Lemma fixedFieldP E {A : {set gal_of E}} a :
a \in E -> reflect (forall x, x \in A -> x a = a) (a \in fixedField A).