paper

Complete Solving for Explicit Evaluation of Gauss Sums in the Index 2 Case

arXiv:0911.5472 · doi:10.1007/s11425-010-3155-z

Abstract

Let be a prime number, for some positive integer , be a positive integer such that , and let $\k$ be a primitive multiplicative character of order over finite field $\fq$. This paper studies the problem of explicit evaluation of Gauss sums in "\textsl{index 2 case}" (i.e. $f=\f{\p(N)}{2}=[\zn:\pp]$, where $\p(\cd)$ is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums $G(\k^\la) (1\laN-1)$ in index 2 case with order being general even integer (i.e. $N=2^{r}\cd N_0$ where are positive integers and is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved.

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