The Reckoning of Certain Quartic and Octic Gauss Sums

In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number A, every nonzero element of G can be written as a difference of two elements of/fin exactly A ways. Throughout the paper, p designates an odd prime. An evaluation of general quartic Gauss sums for/> = 1 (mod 4) can be found in Hasse's book [3, pp. 490-493]. R. J. Evans and the first named author [1] have explicitly evaluated general octic Gauss sums for p = 1 (mod 8). Unfortunately, only a special class of Gauss sums can be evaluated by the method of this note, but we feel that the method's brevity and simplicity are worth noting. For m e {4,8} and p = 1 (mod m), let