Polynomial Dedekind domains with finite residue fields of prime characteristic

Giulio Peruginelli

Vol. 324 (2023), No. 2, 333–351

DOI: 10.2140/pjm.2023.324.333

Abstract

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued polynomials; that is, for each prime $p\in \mathbb{Z}$ there exists a finite subset $E_p$ of transcendental elements over $\mathbb{Q}$ in the absolute integral closure $\overline{{\mathbb{Z}}_p}$ of the ring of $p$-adic integers such that $R=\{f\in \mathbb{Q}[X]\mid f(E_p)\subseteq \overline{{\mathbb{Z}}_p},\text{ for each prime }p\in \mathbb{Z}\}$. Moreover, we prove that the class group of $R$ is isomorphic to a direct sum of a countable family of finitely generated abelian groups. Conversely, any group of this kind is the class group of a Dedekind domain $R$ between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.

Keywords

Dedekind domain, class group, integer-valued polynomials

Mathematical Subject Classification

Primary: 13B25, 13F05, 13F20, 20K99

Milestones

Received: 9 July 2022

Revised: 19 April 2023

Accepted: 27 June 2023

Published: 26 July 2023