proposition: Lemma group_closure_field_exists gT F : classically {Fs : fieldType & {rmorphism F -> Fs} & group_closure_field Fs gT}.
proof: Proof. set n := #|{group gT}|. suffices: classically {Fs : fieldType & {rmorphism F -> Fs} & forall G : {group gT}, enum_rank G < n -> group_splitting_field Fs G}. - apply: classic_bind => [[Fs f splitFs]] _ -> //. by exists Fs => // G; apply: splitFs. elim: (n) => [|i IHi]; first by move=> _ -> //; exists F => //; exists id. apply: classic_bind IHi => [[F' f splitF']]. have [le_n_i _ -> // | lt_i_n] := leqP n i. by exists F' => // G _; apply: splitF'; apply: leq_trans le_n_i. have:= @group_splitting_field_exists _ (enum_val (Ordinal lt_i_n)) F'. apply: classic_bind => [[Fs f' splitFs]] _ -> //. exists Fs => [|G]; first exact: (f' \o f). rewrite ltnS leq_eqVlt -{1}[i]/(val (Ordinal lt_i_n)) val_eqE. case/predU1P=> [defG | ltGi]; first by rewrite -[G]enum_rankK defG. by apply: (extend_group_splitting_field f'); apply: splitF'. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)