Let p be a prime number, N be a positive integer such that gcd(N, p) = 1, q = p f where f is the multiplicative order of p modulo N . Let χ be a primitive multiplicative character of order N over finite field Fq. This paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case" (i.e. [(Z/N Z) * : p ] = 2). Firstly, the classification of the Gauss sums in the index 2 case is presented. Then, the explicit evaluation of Gauss sums G(χ λ ) (1 λ N − 1) in the index 2 case with order N being general even integer (i.e. N = 2 r · N 0 , where r, N 0 are positive integers and N 0 3 is odd) is obtained. Thus, combining with the researches before, the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.
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