End DerivationAlgebra.
Definition separable_element U x := separable_poly (minPoly U x).
Section SeparableElement.
Variables (K : {subfield L}) (x : L).
(* begin hide *)
Let sKxK : (K <= <<K; x>>)%VS := subv_adjoin K x.
Let Kx_x : x \in <<K; x>>%VS := memv_adjoin K x.
(* end hide *)

Lemma separable_elementP :
reflect (exists f, [/\ f \is a polyOver K, root f x & separable_poly f])
(separable_element K x).