proposition: Lemma ker_gal (E : {subfield L}) : ('ker (gal E))%g = kAEndf E.
proof: Proof. apply/setP => g; rewrite !inE kAut1E/= !kAutE subvf andbT subfield_closed. apply/andP/idP => [[gE /gal_eqP galg1]|/kAHomP g_id]. by apply/kAHomP => x xE; have := galg1 _ xE; rewrite gal_id galK// => ->. have gE : (g @: E <= E)%VS by apply/subvP => _/memv_imgP[x ? ->]; rewrite g_id. by split=> //; apply/gal_eqP => x xE; rewrite gal_id galK// g_id. Qed.
imports: HB(structures) mathcomp(all_ssreflect all_fingroup all_algebra) mathcomp(all_solvable all_field) Abel(various char0)