Section FiniteGenField.
Variables (F : finFieldType) (gT : finGroupType) (G : {group gT}) (n' : nat).
Local Notation n := n'.+1.
Variables (rG : mx_representation F G n) (A : 'M[F]_n).
Hypotheses (irrG : mx_irreducible rG) (cGA : centgmx rG A).
Notation FA := (gen_of irrG cGA).
(* This should be [countMixin of FA by <:]*)
Definition gen_countMixin := (sub_countMixin (gen_subType irrG cGA)).
Canonical gen_countType := Eval hnf in CountType FA gen_countMixin.
Canonical gen_subCountType := Eval hnf in [subCountType of FA].
Definition gen_finMixin := [finMixin of FA by <:].
Canonical gen_finType := Eval hnf in FinType FA gen_finMixin.
Canonical gen_subFinType := Eval hnf in [subFinType of FA].
Canonical gen_finZmodType := Eval hnf in [finZmodType of FA].
Canonical gen_baseFinGroupType := Eval hnf in [baseFinGroupType of FA for +%R].
Canonical gen_finGroupType := Eval hnf in [finGroupType of FA for +%R].
Canonical gen_finRingType := Eval hnf in [finRingType of FA].
Canonical gen_finComRingType := Eval hnf in [finComRingType of FA].
Canonical gen_finUnitRingType := Eval hnf in [finUnitRingType of FA].
Canonical gen_finComUnitRingType := Eval hnf in [finComUnitRingType of FA].
Canonical gen_finIdomainType := Eval hnf in [finIdomainType of FA].
Canonical gen_finFieldType := Eval hnf in [finFieldType of FA].

Lemma group_closure_field_exists gT F :
classically {Fs : fieldType & {rmorphism F -> Fs}
& group_closure_field Fs gT}.