Non-unique factorization and principalization in number fields

TL;DR

This paper explores how the class group of a number field quantifies the failure of unique factorization, providing detailed descriptions of irreducible factorizations and their combinatorial structures, with explicit methods involving quadratic forms.

Contribution

It offers a detailed characterization of irreducible factorizations in number fields and connects quadratic forms to explicit factorization descriptions.

Findings

Class group measures non-uniqueness of factorization

Explicit structure of all irreducible factorizations determined

Quadratic forms can explicitly describe factorizations in certain cases

Abstract

We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible factorizations of an element in the ring of integers of a number field, and give a combinatorial description for the number of such factorizations. In certain cases, we show how quadratic forms can explicitly provide all such factorizations.