  We investigate the nilpotence of a kind of circulant matrices $T_{n,m}$ over
field $Z_p$ where $T_{n,m}= \sum_{i = 0}^{m - 1} {S_n^i}$ and $S_n$ is the
fundamental circulant matrix of order $n$. The necessary and sufficient
condition on $n$ and $m$ for determining nilpotence of $T_{n,m}$ over $Z_p$ is
presented. Moreover, we obtain a formula for nilpotent index of $T_{n,m}$ when
the condition is satisfied. As an application, we give a complete solution of a
conjecture of C.Y.Zhang.
