Construction of Universal p-adic Fields

Abstract

In order to be able to define K-valued functions by means of series (mainly power series), we have to assume that K is complete. It turns out that the algebraic closure Q ^a_p is not complete, so we shall consider its completion C _p : This field turns out to be algebraically closed and is a natural domain for the study of “analytic functions.” However, this field is not spherically complete (2.4), and spherical completeness is an indispensable condition for the validity of the analogue of the Hahn-Banach theorem (Ingleton’s theorem (IV.4.7); spherical completeness also appears in (VI.3.6)). This is a reason for enlarging Q ^a_p in a more radical way than just completion, and we shall construct a spherically complete, algebraically closed field Ω _p (containing Q ^a_p and C _p ) having still another convenient property, namely (math). This ensures that all spheres of positive radius in Ω _p are nonempty: B ^<r(a) # B ^<r(a) for all r ≥ 0. In fact, we shall define the big ultrametric extension Ω _p first — using an ultraproduct — and prove all its properties (this method is due to B. Diarra) and then define Cp as the topological closure of Q ^a_p in C _p . This simplifies the proof that C _p is algebraically closed. By a universal p-adic field we mean a complete, algebraically closed extension of Q _p .