Canonical fieldExt_horner_lrmorhism := [lrmorphism of fieldExt_horner].
End Horner.
End FieldExtTheory.
Notation "E :&: F" := (capv_aspace E F) : aspace_scope.
Notation "'C_ E [ x ]" := (capv_aspace E 'C[x]) : aspace_scope.
Notation "'C_ ( E ) [ x ]" := (capv_aspace E 'C[x])
(only parsing) : aspace_scope.
Notation "'C_ E ( V )" := (capv_aspace E 'C(V)) : aspace_scope.
Notation "'C_ ( E ) ( V )" := (capv_aspace E 'C(V))
(only parsing) : aspace_scope.
Notation "E * F" := (prodv_aspace E F) : aspace_scope.
Notation "f @: E" := (aimg_aspace f E) : aspace_scope.
Arguments Fadjoin_idP {F0 L K x}.
Arguments FadjoinP {F0 L K x E}.
Arguments Fadjoin_seqP {F0 L K rs E}.
Arguments polyOver_subvs {F0 L K p}.
Arguments Fadjoin_polyP {F0 L K x v}.
Arguments Fadjoin1_polyP {F0 L x v}.
Arguments minPoly_XsubC {F0 L K x}.
Section MapMinPoly.
Variables (F0 : fieldType) (L rL : fieldExtType F0) (f : 'AHom(L, rL)).
Variables (K : {subfield L}) (x : L).

(* Specifications: *)
Lemma adjoin_degree_aimg : adjoin_degree (f @: K) (f x) = adjoin_degree K x.
Lemma map_minPoly :  map_poly f (minPoly K x) = minPoly (f @: K) (f x).
End MapMinPoly.