NEW EXPLICIT MULTIPLICATIVE RELATIONS BETWEEN GAUSS SUMS

Throughout the paper, for any positive integer k, k will denote a primitive k'th root of unity. Let e be a positive integer, e> 2, and x = e. Let K = Q(e). Let p be a prime such that p 1 mod e and Fp the nite eld with p elements. Let be a generator of the cyclic group Fp. Dene a multiplicative character : Fp! Q(e) by ( ) = e. We extend it by (0) = 0 to a map from Fp to Q(e). For a 2 Z, the Gauss sum (a) is de ned by (a)