The determination of Gauss sums

The first equality in (1.1) is a simple consequence of the fact that the sequence {n}9 1 < n < p — 1, runs through the set of kth power residues (mod/?) exactly k times. The primary purpose of this paper is to survey the present knowledge on the values of the Gauss sums §(k) and G(x), and to convey some of the principal ideas used in their determinations. We also briefly discuss more general Gauss sums. We begin by making some elementary remarks about the values of Gauss sums. It is easily verified by direct multiplication that, for nonprincipal x>

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