On Gaussian Sums Over Finite Fields II

Abstract

Let Fq:=GF(q) be a finite field of order q=ps for some s N, where p is an odd prime, and Fq=F( ), where F=Fp. Let =Fq-{0}= and K:= . Let be a non-trivial irreducible character of the additive group of Fq such that 0 h1,…,hs p-1; (h1,…,hs) (0,…,0) and where ; ; =cos(2 /p)+ isin(2 /p) and be the irreducible character of the multiplicative group with for any i Z. In this paper, it has been determined the values of x and y for the irreducible additive character , by counting square elements satisfying certain conditions in .