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

Abstract

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 $\mathbb{F}_q$. This paper studies the problem of explicit evaluation of Gauss sums G(χ) 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(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.