(* 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.

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).