Elementary properties of power series fields over finite fields

Abstract

In spite of the analogies between Q_p and F_p((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Q_p to the case of F_p((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on F_p((t)). We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also discuss the action of arbitrary polynomials on valued fields.