proposition: 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).
proof: Proof. move=> sKE galEKx Ea; have homKx: kHom K E x by rewrite -gal_kHom. have K_Pa := minPolyOver K a; rewrite -[minPoly K a](fixedPoly_gal _ galEKx) //. by rewrite (kHom_root homKx) ?root_minPoly // (polyOverS (subvP sKE)). Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq div) mathcomp(choice fintype tuple finfun bigop ssralg poly) mathcomp(polydiv finset fingroup morphism quotient perm) mathcomp(action zmodp cyclic matrix mxalgebra vector) mathcomp(falgebra fieldext separable)