Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains

Let $p\in\Z$ be a prime, $\oQp$ a fixed algebraic closure of the field of $p$-adic numbers and $\oZp$ the absolute integral closure of the ring of $p$-adic integers. Given a residually algebraic torsion extension $W$ of $\Z_{(p)}$ to $\Q(X)$, by Kaplansky's characterization of immediate extensions of valued fields, there exists a pseudo-convergent sequence of transcendental type $E=\{s_n\}_{n\in\N}\subset\oQp$ such that $W=\Z_{(p),E}=\{\phi\in\Q(X)\mid\phi(s_n)\in\oZp,\text{ for all sufficiently large }n\in\N\}$. We show here that we may assume that $E$ is stacked, in the sense that, for each $n\in\N$, the residue field (the value group, respectively) of $\oZp\cap\Q_p(s_n)$ is contained in the residue field (the value group, respectively) of $\oZp\cap\Q_p(s_{n+1})$; this property of $E$ allows us to describe the residue field and value group of $W$. In particular, if $W$ is a DVR, then there exists $\alpha$ in the completion $\C_p$ of $\oQp$, $\alpha$ transcendental over $\Q$, such that $W=\Z_{(p),\alpha}=\{\phi\in\Q(X)\mid\phi(\alpha)\in \O_p\}$, where $\O_p$ is the unique local ring of $\C_p$; $\alpha$ belongs to $\oQp$ if and only if the residue field extension $W/M\supseteq\Z/p\Z$ is finite. As an application, we provide a full characterization of the Dedekind domains between $\Z[X]$ and $\Q[X]$.