Extensions of valuations to rational function fields over completions

Arpan Dutta

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4745-4768

Abstract: Given a valued field (K,v)$(K,v)$ and its completion (K̂,v)$(\widehat{K},v)$, we study the set of all possible extensions of v to K̂(X)$\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension (K(X)|K,v)$(K(X)|K,v)$. 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 (K̂,v)$(\widehat{K},v)$. We also give necessary and sufficient conditions for (K(X),v)$(K(X),v)$ to be dense in (K̂(X),v)$(\widehat{K}(X),v)$.

Date: 2023
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https://doi.org/10.1002/mana.202200178

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