Henselian elements

Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension (F|L,v), if F is contained in the absolute inertia field of L, then the valuation ring OF of (F,v) is generated as an OL-algebra by henselian elements. Moreover, we give a list of equivalent conditions under which OF is generated over OL by finitely many henselian elements. We prove that if the chain of prime ideals of OL is well-ordered, then these conditions are satisfied. We give an example of a finite valued inertial extension (F|L,v) for which OF is not a finitely generated OL-algebra. We also present a theorem that relates the problem of local uniformization to the theory of henselian elements.