Section ExtendDerivation.
Variable D : 'End(L).
Let Dx E := - (map_poly D (minPoly E x)).[x] / ((minPoly E x)^`()).[x].
Fact extendDerivation_subproof E (adjEx := Fadjoin_poly E x) :
let body y (p := adjEx y) := (map_poly D p).[x] + p^`().[x] * Dx E in
linear body.
Definition extendDerivation E : 'End(L) :=
linfun (Linear (extendDerivation_subproof E)).
Hypothesis derD : Derivation K D.
End ExtendDerivation.
End SeparableElement.
Arguments separable_elementP {K x}.
Definition purely_inseparable_element U x :=
x ^+ ex_minn (separable_exponent <<U>> x) \in U.

Lemma purely_inseparable_elementP {K x} :
reflect (exists2 n, [char L].-nat n & x ^+ n \in K)
(purely_inseparable_element K x).