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).

(* Specifications: *)
Lemma integral_algebraic u : algebraicOver FtoE u <-> integralOver FtoE u.
Lemma algebraic_id a : algebraicOver FtoE (FtoE a).
Lemma algebraic_sub x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x - y).
Lemma algebraic_inv u : algebraicOver FtoE u -> algebraicOver FtoE u^-1.
Lemma algebraic_div x y :
algebraicOver FtoE x -> algebraicOver FtoE y -> algebraicOver FtoE (x / y).