Section PolyXY_Field.
Variables (F E : fieldType) (FtoE : {rmorphism F -> E}).
Local Notation pFtoE := (map_poly (GRing.RMorphism.apply FtoE)).

Lemma algebraic_root_polyXY x y :
(let pEx p := (p ^ map_poly FtoE).[x%:P] in
exists2 p, pEx p != 0 & root (pEx p) y) ->
algebraicOver FtoE x -> algebraicOver FtoE y.