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

$$W={\mathbb{Z}}_{(p),E}=\{\varphi \in \mathbb{Q}(X)\mid \varphi (s_n)\in \overline{{\mathbb{Z}}_p}\text{for all sufficiently large }n\in \mathbb{N}\}.$$

We show here that we may assume that $E$ is stacked, in the sense that, for each $n\in \mathbb{N}$, the residue field (resp. the value group) of $\overline{{\mathbb{Z}}_p}\cap {\mathbb{Q}}_p(s_n)$ is contained in the residue field (resp. the value group) 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 ${ℂ}_p$ of $[t]\overline{{\mathbb{Q}}_p}$, $\alpha$ transcendental over $\mathbb{Q}$, such that $W={\mathbb{Z}}_{(p),\alpha }=\{\varphi \in \mathbb{Q}(X)\mid \varphi (\alpha )\in {\mathbb{𝕆}}_p\}$, where ${\mathbb{𝕆}}_p$ is the unique local ring of ${ℂ}_p$; $\alpha$ belongs to $[t]\overline{{\mathbb{Q}}_p}$ 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 $\mathbb{Q}[X]$.