Transcendental extensions of a valuation domain of rank one

Let $V$ be a valuation domain of rank one and quotient field $K$. Let $abK$ be a fixed algebraic closure of the $v$-adic completion $K$ of $K$ and let $abV$ be the integral closure of $V$ in $abK$. 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 $alphainabKcup{infty}$, namely, the valuation domains $W=W_{alpha}={arphiin K(X) mid arphi(alpha)inabV}$. If $V$ is discrete and $piin V$ 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 $egeq 1$, where $M$ is the maximal ideal of $W$. In general, for $alpha,etainabK$ we have $W_{alpha}=W_{eta}$ if and only if $alpha$ and $eta$ are conjugated over $K$. Finally, we show that the set $Pirr$ of irreducible polynomials over $K$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space ${W_{alpha} mid alphainabK}$ endowed with the Zariski topology