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

TL;DR

This paper provides a complete explicit evaluation of Gauss sums in the index 2 case for finite fields, covering general even orders, and classifies these sums comprehensively.

Contribution

It offers the first complete solution for explicit Gauss sum evaluation in the index 2 case, including classification and evaluation for general even orders.

Findings

Classification of Gauss sums in index 2 case

Explicit evaluation formulas for even order N

Complete resolution of the explicit evaluation problem

Abstract

Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let $\k$ be a primitive multiplicative character of order $N$ 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 $N$ being general even integer (i.e. $N=2^{r}\cd N_0$ where $r,N_0$ are positive integers and $N_03$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved.