Determination Of Gauss Sum In Case Of Index 4

Gauss sums are one of the most important and fundamental objects and tools in number theory. The explicit evaluation of Gauss sums is a very important but very difficult problem, which has not only theoretical value in number theory and arithmetical geometry, but also important practical applications in computer science, information theory, combinatorics and experimental designs.From C. F. Gauss to contemporary mathematicians, a lot of investigations have been done on this problem. But there are only a few situations that one can clearly determine the value of Gauss sums. Presently, the researches have been done in the following two directions: the first one is the case of Gauss sums with low orders. In this case, one can evaluate Gauss sums by the comparatively simpler arithmetic properties of the number fields with low degrees over Q. For the other direction, people have studied Gauss sums by analyzing the arithmetic properties of cyclotomic fields and their subfields with Galois' Theory. When the index r is a small natural number, one can evaluate Gauss sums by the arithmetic properties of the number fields with degree r over Q. Recently, the explicit formulas of Gauss sums have been determined in the case of "index 2" .In this dissertation we present explicit formulas for Gauss sums G(x) in the case of "index 4" . According to N, the order of the multiplicative character x of G(x), being odd or even, we discuss two different cases.When N is odd, Gauss sums G(x) belong to a certain imaginary quartic number field K. We firstly obtain the decomposition of G(x) as a product of prime ideals in integral ring of K by the Stickelberger Relation. Then we get two different types of explicit formula of G(x) according to the Galois group of K being a cyclic group with order 4 or a direct product of two cyclic groups with order 2. For the cyclic case, the value of G(x) is determined by the integral solutions of a certain quadratic Diophantine equations and related to the relative ideal class number of K. For the non-cyclic case, there are more subcases, however, the formulas of G(x) are relatively simpler and related to the ideal class numbers of the two imaginary quadratic subfields of K. In thecourse of evaluation, we have obtained two different methods to determinate the sign of G(x), and the second method is more explicit, brief and unified.When N is even, Gauss sums G(x) just partially belong to a certain imaginary quartic number field K. And the calculational feasibility of the part outside K owes to the calculational feasibility of Gauss sums with lower orders over F_p. When N is the power of 2, we have also given the explicit formulas of G(x), both in the cyclic case and non-cyclic case.