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

Knight, Lucy  ; Stasinski, Alexander 

Glasgow mathematical journal, Tome 66 (2024) no. 1, pp. 88-103

Résumé

For a principal ideal domain $A$, the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\textrm{M}_{n}(A)$ 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 The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

Détail

DOI : 10.1017/S0017089523000356

Mots-clés : Matrices of integers, matrices over principal ideal domains, Dedekind domains, ideal class groups, prime ideals of degree one