Computing the ideal class monoid of an order

TL;DR

This paper extends existing algorithms to compute the isomorphism classes of all ideals, including non-invertible ones, in orders of number fields, and generalizes a theorem relating matrices and lattice classes.

Contribution

It introduces methods to compute all ideal classes in an order, including non-invertible ideals, and generalizes a theorem connecting matrices and lattice classes.

Findings

Algorithms for non-invertible ideal classes are developed.

Extended the Latimer-MacDuffee theorem to broader settings.

Provides explicit descriptions of ideal class bijections.

Abstract

There are well known algorithms to compute the class group of the maximal order $\mathcal{O}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order $R$ in a finite product of number fields $K$. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain $\mathbb{Q}$-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.