Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

TL;DR

This paper explores the correspondence between matrix similarity classes over principal ideal domains and ideal classes, showing that under certain conditions, each class has a representative close to a companion matrix, with implications for algebraic number theory.

Contribution

It proves that when the order is maximal, each similarity class has a representative close to a companion matrix, extending the Latimer--MacDuffee correspondence.

Findings

Every ideal class contains an ideal of degree one when the order is maximal.

Similarity classes corresponding to degree-one ideals have representatives close to companion matrices.

The result relies on an unpublished lemma by Lenstra about ideal classes in maximal orders.

Abstract

For a principal ideal domain $A$, the Latimer--MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$ is maximal (i.e., integrally closed, i.e., a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.