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Representatives of similarity classes of matrices over PIDs corresponding to ideal classes
- Part of
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- Journal:
- Glasgow Mathematical Journal / Volume 66 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 18 October 2023, pp. 88-103
- Print publication:
- January 2024
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- Article
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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 The second step is a previously unpublished result due to Lenstra that implies that when 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.
