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

Lemma div_annihilantP (p q : {poly E}) (x y : E) :
p != 0 -> q != 0 -> y != 0 -> p.[x] = 0 -> q.[y] = 0 ->
(div_annihilant p q).[x / y] = 0.