Power Series and p-adic Algebraic Closures

In a previous paper [3], the author gave an explicit description of the algebraic closure of the power series field over a field of characteristic p > 0, in terms of certain “generalized power series”. The purpose of the present paper is to extend this work to mixed characteristic. Specifically, we give an analogous description of the algebraic closure of the Witt ring W (K) of an algebraically closed field K of characteristic p. In place of generalized power series, we use “generalized p-adic series” as introduced by Poonen [6]. (Note: here and throughout, the “algebraic closure” of a domain refers to the integral closure of the domain in the algebraic closure of its function field.) Our approach is to relate the algebraic closure of W (K) to the algebraic closure of K[[t]] via the Witt ring of the latter. We first exhibit a surjection of W (K[[t]]) onto W (K) ∧ (the wedge denotes p-adic closure) in which t maps to p. (The wedge denotes p-adic completion.) Using the results of [3], we then give explicit presentations of W (K[[t]]) and of W (K) ∧ . Our approach makes use of two notions which occur in [3] but might otherwise be unfamiliar to the reader: generalized power series, described in Section 2, and twist-recurrent sequences (or linearized recurrent sequences), described in Section 3.