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

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 $\text{k}$ be a primitive multiplicative character of order $N$ over finite field $\text{fq}$. This paper studies the problem of explicit evaluation of Gauss sums in “\textsl{index 2 case}” (i.e. $f=\text{f}\text{p}(N)2=[\text{zn}:\text{pp}]$, where $\text{p}(\text{cd})$ is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums $G({\text{k}}^{\text{la}})(1\text{laN}-1)$ in index 2 case with order $N$ being general even integer (i.e. $N=2^r\text{cd}{N}_0$ where $r,N_0$ are positive integers and $N_{0}3$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved.