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- W4229031812 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 The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one." @default.
- W4229031812 created "2022-05-08" @default.
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- W4229031812 date "2022-05-04" @default.
- W4229031812 modified "2023-09-23" @default.
- W4229031812 title "Representatives of similarity classes of matrices over PIDs corresponding to ideal classes" @default.
- W4229031812 doi "https://doi.org/10.48550/arxiv.2205.02094" @default.
- W4229031812 hasPublicationYear "2022" @default.
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