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arXiv · 2505.09635

On ideal class groups of totally degenerate number rings

Ruben Hambardzumyan·Mihran Papikian
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Abstract

Let $\chi(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(\chi(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $\chi(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $\chi(x)$ is $2$ or $3$.

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Ruben Hambardzumyan, Mihran Papikian. 2025-05-04. On ideal class groups of totally degenerate number rings. https://arxiv.org/abs/2505.09635

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