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

Lemma root_annihilant x p (pEx := (p ^ pFtoE).[x%:P]) :
pEx != 0 -> algebraicOver FtoE x ->
exists2 r : {poly F}, r != 0 & forall y, root pEx y -> root (r ^ FtoE) y.