On a class of diagonal equations over finite fields

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\cdots +x_n^{2^m}=0$ over a finite field of characteristic $p\equiv \pm 3(\mathrm{mod}8)$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of p.