Transcendental extensions of a valuation domain of rank one

TL;DR

This paper characterizes a class of valuation domains over a rational function field extending a rank-one valuation domain, linking algebraic properties with topological structures via Krasner's ultrametric.

Contribution

It describes valuation domains over $K(X)$ indexed by algebraic elements, relating their algebraic structure to topological properties and conjugacy classes over the completion.

Findings

Valuation domains $W_ ext{ extalpha}$ are characterized by elements $ extalpha$ in the algebraic closure.

The set of valuation domains is homeomorphic to the space of irreducible polynomials over $ exthat{K}$.

Explicit conditions for valuation domains when $V$ is discrete and uniformizer exists.

Abstract

Let $V$ be a valuation domain of rank one and quotient field $K$. Let $\overline{\hat{K}}$ be a fixed algebraic closure of the $v$-adic completion $\hat K$ of $K$ and let $\overline{\hat{V}}$ be the integral closure of $\hat V$ in $\overline{\hat{K}}$. We describe a relevant class of valuation domains $W$ of the field of rational functions $K(X)$ which lie over $V$, which are indexed by the elements $\alpha\in\overline{\hat{K}}\cup\{\infty\}$, namely, $W=W_{\alpha}=\{\varphi\in K(X) \mid \varphi(\alpha)\in\overline{\hat{V}}\}$. If $V$ is discrete and $\pi\in V$ is a uniformizer, then a valuation domain $W$ of $K(X)$ is of this form if and only if the residue field degree $[W/M:V/P]$ is finite and $\pi W=M^e$, for some $e\geq 1$, where $M$ is the maximal ideal of $W$. In general, for $\alpha,\beta\in\overline{\hat{K}}$ we have $W_{\alpha}=W_{\beta}$ if and only if $\alpha$ and $\beta$ are conjugated over $\hat K$. Finally, we show that the set $\mathcal{P}^{{\rm irr}}$ of irreducible polynomials over $\hat K$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space $\{W_{\alpha} \mid \alpha\in\overline{\hat{K}}\}$ endowed with the Zariski topology.