Elliptic Dedekind Domains Revisited

TL;DR

This paper proves that every abelian group can be realized as the class group of an elliptic Dedekind domain, providing new constructions and simpler proofs for classical theorems in algebraic number theory.

Contribution

It answers a longstanding question by showing all abelian groups are class groups of elliptic Dedekind domains, with explicit constructions involving quadratic fields.

Findings

Every abelian group is isomorphic to a class group of an elliptic Dedekind domain.

Constructs such domains as integral closures of PIDs in quadratic fields.

Provides simpler proofs of classical theorems by Claborn and Leedham-Green.

Abstract

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and -- we feel -- simpler proofs of theorems of L. Claborn and C.R. Leedham-Green.