  In this paper, we address the problem for enumerating the number of finite
field elements with prescribed trace and co-trace in case of arbitrary
characteristic $p$. We show that this problem can be reduced to solving a
system of $p-1$ linear equations with matrix of coefficients a slight
modification of circulant matrix formed by the Kloosterman sums over the field
$\MF_{p}$. The presented approach is illustrated in the cases of characteristic
$p = 2,3$ and $5$.
