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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.09635 |
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| _version_ | 1866909929687744512 |
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| author | Hambardzumyan, Ruben Papikian, Mihran |
| author_facet | Hambardzumyan, Ruben Papikian, Mihran |
| contents | Let $χ(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(χ(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $χ(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $χ(x)$ is $2$ or $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On ideal class groups of totally degenerate number rings Hambardzumyan, Ruben Papikian, Mihran Number Theory 11R29, 11R54, 15B36 Let $χ(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(χ(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $χ(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $χ(x)$ is $2$ or $3$. |
| title | On ideal class groups of totally degenerate number rings |
| topic | Number Theory 11R29, 11R54, 15B36 |
| url | https://arxiv.org/abs/2505.09635 |