Transcendental extensions of a valuation domain of rank one

Abstract:

Let $V$ be a valuation domain of rank one, maximal ideal $P$ and quotient field $K$. Let $\overline {\widehat {K}}$ be a fixed algebraic closure of the $v$-adic completion $\widehat {K}$ of $K$ and let $\overline {\widehat {V}}$ be the integral closure of $\widehat {V}$ in $\overline {\widehat {K}}$. We describe a relevant class of valuation domains $W$ of the field of rational functions $K(X)$ lying over $V$, which are indexed by the elements $\alpha \in \overline {\widehat {K}}\cup \{\infty \}$, namely, the valuation domains $W=W_{\alpha }=\{\varphi \in K(X) \mid \varphi (\alpha )\in \overline {\widehat {V}}\}$. 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 $\alpha ,\beta \in \overline {\widehat {K}}$ we have $W_{\alpha }=W_{\beta }$ if and only if $\alpha$ and $\beta$ are conjugate over $\widehat {K}$. Finally, we show that the set ${\mathcal {P}^{\textrm {irr}}}$ of irreducible polynomials over $\widehat {K}$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space $\{W_{\alpha } \mid \alpha \in \overline {\widehat {K}}\}$ endowed with the Zariski topology.