  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.
