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arXiv · 2604.25461

Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$

Hayaki Kudo·Yuto Nogata
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Abstract

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 $r=3$.

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Hayaki Kudo, Yuto Nogata. 2026-04-28. Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$. https://arxiv.org/abs/2604.25461

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