Gauss Sums Of Index Four

Gauss sums is one of important objects in number theory. To calculate the value of Gauss sums is one of important and difficult problems which has not only theoretical meaning in number theory and arithmetical geometry, but also practical applications in computer science, information theory and statistical designs.The first computational result on Gauss sums is given by Gauss himself around 1800 and is applied to the famous Gauss quadratic reciprocity law. After this, the values of m-th Gauss sums have been determined for small m(= 3,4, ···, 12) by using arithmetic properties of cyclotomic fields Q(sm)- Recently the formulas of Gauss sums are determined in "self-conjugate" and "index 2" cases in which the Gauss sums belongs to the rational number field and certain imaginary quadratic field respectively.In this thesis we present explicit formulas on Gauss sums G(x) in " index 4 " case in which G(x) belongs to a certain imaginary quadratic number field K. Firstly we obtain the decomposition of G{x) as a product of prime ideals in K by the Stickelberger theorem. Then we get formula of G(x) with two different types according to K being cyclic or non-cyclic. For cyclic case, the value of G(x) is determined by the integral solutions of a certain quadratic Diophantin equations and related to the relative class number of K. For non-cyclic case the formula of G(x) is relatively simpler and related to the class number of two imaginary quadratic subfields of K.