Dirichlet character modulo such that and are relatively prime, then The relation among , , and when and are of the same modulus (and is primitive) is measured by the Jacobi sum . Specifically, Further properties Gauss sums can be used to prove quadratic reciprocity, cubic reciprocity and quartic reciprocity Gauss sums can be used to calculate the number of solutions of polynomial equations over finite fields, and thus can be used to calculate certain zeta functions See also Chowla–Mordell theorem Elliptic Gauss sum Gaussian period Hasse–Davenport relation Jacobi sum Stickelberger's theorem Quadratic Gauss sum Kummer sum References Section 3.4 of Cyclotomic fields