Extensions of valuations to rational function fields over completions

Abstract

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

arXiv commentFinal version, to appear in Mathematische Nachrichten