(* Standard mathematical notation for 'Gal(E / K) puts the larger field first.*)
Local Notation "''Gal' ( V / U )" := (galoisG V U) : group_scope.
Canonical galoisG_group E U := Eval hnf in [group of (galoisG E U)].
Local Notation "''Gal' ( V / U )" := (galoisG_group V U) : Group_scope.
Section Automorphism.

(* Specifications: *)
Lemma gal_cap U V : 'Gal(V / U) = 'Gal(V / U :&: V).
Lemma gal_kHom K E x : (K <= E)%VS -> (x \in 'Gal(E / K)) = kHom K E x.
Lemma fixed_gal K E x a :
(K <= E)%VS -> x \in 'Gal(E / K) -> a \in K -> x a = a.
Lemma root_minPoly_gal K E x a :
(K <= E)%VS -> x \in 'Gal(E / K) -> a \in E -> root (minPoly K a) (x a).