Extensions of valuations to rational function fields over completions

Abstract

Given a valued field (K, v) and its completion ( K, v), we study the set of all possible extensions of v to K(X). We show that any such extension is closely connected with the underlying subextension (K(X) |K, v). The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of ( K, v). We also give necessary and sufficient conditions for (K(X), v) to be dense in ( K(X), v).In this article, we undertake a systematic study of the extension ( K(X)| K, v), where K denotes the completion of the valued field (K, v). Our investigation shows that the extension ( K(X)| K, v) is intimately connected with the underlying extension (K(X) |K, v). The connection between these extensions is further explored in detail in terms of the aforementioned objects, viz. minimal pairs, key polynomials and implicit constant fields. Our first important result is the following: Theorem 1.1. Let (K, v) be a valued field, ( K, v) be its completion and (K(X)|K, w) be an extension of valued fields. Fix an extension v of v to the algebraic closure K of K. Fix a common extension w of w and v to K(X), where v := v| K . Then there exists a common extension of v