Rings of small rank over a Dedekind domain and their ideals

TL;DR

This paper generalizes Bhargava's parametrizations of rings and their ideals from the integers to arbitrary Dedekind domains, enabling new insights into the structure and distribution of low-degree extensions of number fields.

Contribution

It extends Bhargava's parametrizations to Dedekind domains, providing bijections for quadratic, cubic, and quartic rings and an analogue of Gauss composition.

Findings

Parametrizations of quadratic, cubic, and quartic rings over Dedekind domains.

An analogue of Gauss composition law for Dedekind domains.

Potential applications to the distribution of number field extensions.

Abstract

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.