二次互倒律与二次Gauss和的计算

Calculation on Quadratic Reciprocity Law and the Quadratic Gauss Sums

二次互倒律是初等数论中最著名的一个定理 ,它是由法国数学家Legendre等人发现的 ,德国数学家Gauss把这个结果称为是数论的酵母 ,并且首先给出了它的完全的证明。其后世界上多位数学家对互倒律作了重要的推广。而在互倒律的发展和证明过程中 ,Gauss和曾经起过重要的作用。另一方面 ,二次Gauss和又是一种特殊的特征和 ,而特征和是数论中的一个重要工具 ,它在数论的一系列重要问题的研究中有着广泛的应用。利用线性代数的知识 ,作出一个迹为二次Gauss和的n阶矩阵 ,根据线性代数中矩阵的迹等于其所有特征值之和这一基本性质 ,通过求出矩阵所有的特征值来求得二次Gauss和的值 ,从而给出了一种新的计算二次Gauss和的方法。

Quadratic reciprocity law is the most famous theorem in the elementary number theory, found by famous French Mathematician Legendre. The famous German Mathematician Gauss called it 'the yeast of number theory' and gave the first complete proof for it, which have been developed afterwards by many mathematicians all over the world. In the process of the development and the proof of the quadratic reciprocity law, the quadratic Gauss sums had played very important role. On the other hand, quadratic Gauss sums are a special kind of character sums which are very important tools in the study of the number theory with many applications in a series of important problems. In this paper, the authors have given a new method of calculating the quadratic Gauss sums by using the method from the linear algebra. The main idea is to construct a matrix of which the trace is just the quadratic Gauss sums thus that could be calculated if the corresponding characteristic values of the matrix could be given by the linear algebra method.