Approximating DVRs by elements of bounded ramification

Abstract

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 $\overline{\widehat{K}}$ 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 $V_α=\{ϕ\in K(X)\mid v(ϕ(α))\geq0\}$, for an element $α$ of $\mathbb{K}$ transcendental over $K$, and that $α$ is algebraic over $\widehat K$ if and only if the residue field extension is finite. Not every such $V_α$ is a DVR, however, and we characterize the $α\in\mathbb{K}$ for which $V_α$ is a DVR: they are the elements which can be approximated by algebraic elements in $\overline{\widehat{K}}$ 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 $α\in\mathbb K$ for which $V_α$ is a DVR is a subfield of $\mathbb K$, which sits properly between $\overline{\widehat{K}}$ and $\mathbb K$, and corresponds to those elements $α$ for which the value group of $\widehat K(α)$ is discrete.