Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains

Abstract

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