  In an earlier work, the authors have determined all possible weights $n$ for
which there exists a vanishing sum $\zeta_1+\cdots +\zeta_n=0$ of $m$th roots
of unity $\zeta_i$ in characteristic 0. In this paper, the same problem is
studied in finite fields of characteristic $p$. For given $m$ and $p$, results
are obtained on integers $n_0$ such that all integers $n\geq n_0$ are in the
``weight set'' $W_p(m)$. The main result $(1.3)$ in this paper guarantees,
under suitable conditions, the existence of solutions of $x_1^d+\cdots+x_n^d=0$
with all coordinates not equal to zero over a finite field.
