Polynomial Dedekind domains with finite residue fields of prime characteristic

TL;DR

This paper characterizes certain Dedekind domains between polynomial rings over integers and rationals with finite residue fields of prime characteristic, showing they are generalized integer-valued polynomial rings and analyzing their class groups.

Contribution

It provides a complete description of Dedekind domains with finite prime characteristic residue fields as generalized integer-valued polynomial rings and characterizes their class groups.

Findings

Dedekind domains are generalized rings of integer-valued polynomials.

Class groups are isomorphic to direct sums of finitely generated abelian groups.

Any such class group can be realized by a Dedekind domain between Z[X] and Q[X].

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 $R=\{f\in\mathbb Q[X]\mid f(E_p)\subseteq \overline{\mathbb Z_p}, \forall \text{ prime }p\in\mathbb Z\}$. 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 $\mathbb Q[X]$.