Extensions of valuations to rational function fields over completions

Abstract

Given a valued field (Figure presented.) and its completion (Figure presented.), we study the set of all possible extensions of v to (Figure presented.). We show that any such extension is closely connected with the underlying subextension (Figure presented.). 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 (Figure presented.). We also give necessary and sufficient conditions for (Figure presented.) to be dense in (Figure presented.). � 2023 Wiley-VCH GmbH.