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

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 $(a,b{)}_{q-1}\le \lceil \frac{k}{2}\rceil$ 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 $r=3$.