【作者】 罗世新；

【导师】 冯克勤；

【作者基本信息】 清华大学 ， 基础数学， 2004， 硕士

【摘要】 高斯和是数论中一个重要的研究对象。高斯和的计算是一个重要和困难的问题,不仅在数论和算数几何中具有理论价值,而且在计算机科学、信息科学和试验设计等方面有实际的应用。 高斯和的第一个计算结果是由高斯于1800年给出(二次高斯和),用来研究他的著名的二次互反律。继高斯之后,人们用代数数论对于m较小情形计算出m次高斯和。近年来,人们对于“自共轭”情形和“指数2”情形算出高斯和。对于这两种情形,高斯和的值分别属于有理数域和虚二次域。 本文对于“指数4”情形给出高斯和的计算公式。这时它属于某个虚4次阿贝尔数域K。我们首先用Stickelberger定理给出高斯和在K中的素理想分解。然后按K为循环域和非循环域两种不同情形,得到不同类型的计算公式。对于循环情形,高斯和由一个二次方程组的整数解所决定,并且与K的相对理想类数有关。对于非循环情形,计算公式较为简单,并与K的两个虚二次子域的理想类数有关。更多还原

【Abstract】 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.更多还原