Counting principal ideals of small norm in the simplest cubic fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929673628286976 |
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| author | Zindulka, Mikuláš |
| author_facet | Zindulka, Mikuláš |
| contents | We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting principal ideals of small norm in the simplest cubic fields Zindulka, Mikuláš Number Theory 11R16, 11R80 We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm. |
| title | Counting principal ideals of small norm in the simplest cubic fields |
| topic | Number Theory 11R16, 11R80 |
| url | https://arxiv.org/abs/2501.06889 |