Extensions of valuations to rational function fields over completions

TL;DR

This paper explores how valuations extend from a valued field to its completion's rational function field, revealing structural connections and conditions for density and ramification properties.

Contribution

It characterizes extensions of valuations to rational function fields over completions using key tools like minimal pairs and pseudo-Cauchy sequences, providing new structural insights.

Findings

Extensions are closely linked with subextensions $(K(X)|K,v)$

Conditions for density of $(K(X),v)$ in $( ext{completion}(K),v)$ are established

Strong ramification properties of the completion are derived

Abstract

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of $v$ to $\widehat{K}(X)$. 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 $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.