End IntegralOverComRing.
Section IntegralOverField.
Variables (F E : fieldType) (FtoE : {rmorphism F -> E}).
Definition algebraicOver (fFtoE : F -> E) u :=
exists2 p, p != 0 & root (map_poly fFtoE p) u.
Notation mk_mon p := ((lead_coef p)^-1 *: p).

Lemma algebraic_sub x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x - y).