Approximating DVRs by elements of bounded ramification

Let $V$ be a DVR with quotient field $K$ and perfect residue field, $v$ be the valuation on $K$ associated with $V$, $\widehat{K}$ be the completion of $K$, and $\mathbb{K}$ be the completion of an algebraic closure $\bar{\widehat{K}}$ of $\widehat{K}$. We show that a DVR of the rational function field $K(X)$ which is a residually algebraic extension of $V$ is necessarily of the form $V_{\alpha }=\{\phi \in K(X)\mid v(\phi (\alpha ))\ge 0\}$, for an element $\alpha$ of $\mathbb{K}$ transcendental over $K$, and that $\alpha$ is algebraic over $\widehat{K}$ if and only if the residue field extension is finite. Not every such $V_{\alpha }$ is a DVR, however, and we characterize the $\alpha \in \mathbb{K}$ for which $V_{\alpha }$ is a DVR: they are the elements which can be approximated by algebraic elements in $\bar{\widehat{K}}$ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of $V$ to $K(X)$ which are DVRs and residually algebraic over $V$, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of $\alpha \in \mathbb{K}$ for which $V_{\alpha }$ is a DVR is a subfield of $\mathbb{K}$, which sits properly between $\bar{\widehat{K}}$ and $\mathbb{K}$, and corresponds to those elements $\alpha$ for which the value group of $\widehat{K}(\alpha )$ is discrete.