  In this article, we give upper bounds for cyclotomic numbers of order e over
a finite field with q elements, where e is a divisor of q-1. In particular, we
show that under certain assumptions, cyclotomic numbers are at most
$\lceil\frac{k}{2}\rceil$, and the cyclotomic number (0,0) is at most
$\lceil\frac{k}{2}\rceil-1$, where k=(q-1)/e. These results are obtained by
using a known formula for the determinant of a matrix whose entries are
binomial coefficients.
