Complete solving of explicit evaluation of Gauss sums in the index 2 case

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. be a primitive multiplicative character of order N over finite field F(q). This paper studies the problem of explicit evaluation of Gauss sums G(chi) in the "index 2 case" (i.e. [(Z/NZ)* : < p >] = 2). Firstly, the classification of the Gauss sums in the index 2 case is presented. Then, the explicit evaluation of Gauss sums G(chi(lambda)) ( 1 <= lambda <= 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.