Explicit counting of ideals in number fields of arbitrary degree
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917426574131200 |
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| author | Fehnker, Anton |
| author_facet | Fehnker, Anton |
| contents | We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term and avoid terms on the order of $n^{n^2}$. We do this by studying fundamental domains for the action of multiplying with units of infinite order in Minkowski space. With some lattice theory we show that one can make different choices for such a fundamental domain, which yield a smaller error, especially when the degree of the field extension is large. We also adapt Schmidt's partition trick to this generalised setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit counting of ideals in number fields of arbitrary degree Fehnker, Anton Number Theory 11R47, 11N45, 11H16 We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term and avoid terms on the order of $n^{n^2}$. We do this by studying fundamental domains for the action of multiplying with units of infinite order in Minkowski space. With some lattice theory we show that one can make different choices for such a fundamental domain, which yield a smaller error, especially when the degree of the field extension is large. We also adapt Schmidt's partition trick to this generalised setting. |
| title | Explicit counting of ideals in number fields of arbitrary degree |
| topic | Number Theory 11R47, 11N45, 11H16 |
| url | https://arxiv.org/abs/2604.19681 |