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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.25461 |
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| _version_ | 1866915962979090432 |
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| author | Kudo, Hayaki Nogata, Yuto |
| author_facet | Kudo, Hayaki Nogata, Yuto |
| contents | Let $q=p^n$, $r\in \mathbb{Z}_{\ge 2}$, $e=q-1$, and $k=\frac{q^r-1}{e}$. In this paper, we study the cyclotomic numbers $(a,b)_{q-1}$ over $\mathbb{F}_{q^r}$. We prove that $(a,b)_{q-1}\le \left\lceil \frac{k}{2}\right\rceil$ for all $0\le a,b\le q-2$ except when $q=2$ and $r\ge 3$. We also give sharper bounds for prime values of $r$, especially for $r=2$ and $r=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$ Kudo, Hayaki Nogata, Yuto Number Theory 11T22, 12E20, 11T24 Let $q=p^n$, $r\in \mathbb{Z}_{\ge 2}$, $e=q-1$, and $k=\frac{q^r-1}{e}$. In this paper, we study the cyclotomic numbers $(a,b)_{q-1}$ over $\mathbb{F}_{q^r}$. We prove that $(a,b)_{q-1}\le \left\lceil \frac{k}{2}\right\rceil$ for all $0\le a,b\le q-2$ except when $q=2$ and $r\ge 3$. We also give sharper bounds for prime values of $r$, especially for $r=2$ and $r=3$. |
| title | Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$ |
| topic | Number Theory 11T22, 12E20, 11T24 |
| url | https://arxiv.org/abs/2604.25461 |