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

Knight, Lucy; Stasinski, Alexander

Abstract

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 The second step is a previously unpublished result due to Lenstra that implies that when A[x]/(f(x)) is maximal, every ideal class contains an ideal of degree one.

Citation

Knight, L., & Stasinski, A. (2023). Representatives of similarity classes of matrices over PIDs corresponding to ideal classes. Glasgow Mathematical Journal, 66(1), 88-103. https://doi.org/10.1017/s0017089523000356

• Journal Article Type: Article
• Acceptance Date: Sep 20, 2023
• Online Publication Date: Oct 18, 2023
• Publication Date: Oct 18, 2023
• Deposit Date: Dec 12, 2023
• Publicly Available Date: Dec 12, 2023
• Journal: Glasgow Mathematical Journal
• Print ISSN: 0017-0895
• Electronic ISSN: 1469-509X
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 66
• Issue: 1
• Pages: 88-103
• DOI: https://doi.org/10.1017/s0017089523000356
• Keywords: General Mathematics
• Public URL: https://durham-repository.worktribe.com/output/2023057