End Automorphism.

(* Specifications: *)
Lemma gal_adjoin_eq K a x y :
x \in 'Gal(<<K; a>> / K) -> y \in 'Gal(<<K; a>> / K) ->
(x == y) = (x a == y a).
Lemma galS K M E : (K <= M)%VS -> 'Gal(E / M) \subset 'Gal(E / K).
Lemma gal_conjg K E x : 'Gal(E / K) :^ x = 'Gal(E / x @: K).
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).
Lemma mem_fixedFieldP E (A : {set gal_of E}) a :
a \in fixedField A -> a \in E /\ (forall x, x \in A -> x a = a).