Definition purely_inseparable_element U x :=
x ^+ ex_minn (separable_exponent <<U>> x) \in U.

(* Specifications: *)
Lemma purely_inseparable_elementP {K x} :
reflect (exists2 n, [char L].-nat n & x ^+ n \in K)
(purely_inseparable_element K x).
Lemma separable_inseparable_element K x :
separable_element K x && purely_inseparable_element K x = (x \in K).
Lemma base_inseparable K x : x \in K -> purely_inseparable_element K x.
Section PrimitiveElementTheorem.