Definition separable_element U x := separable_poly (minPoly U x).

(* Specifications: *)
Lemma separable_elementS K E x : (K <= E)%VS -> separable_element K x -> separable_element E x.
Lemma sub_inseparable K E x : (K <= E)%VS -> purely_inseparable_element K x -> purely_inseparable_element E x.