  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.
