Construction of abelian equations and study of Gaussian sums

本课题的主要研究课题是逆伽罗瓦理论的构造性问题，目的是建立系统的构造阿贝尔方程组族的方法，这是数论中的核心问题之一。在这项研究工作中，我们的兴趣集中在循环方程的情况下。我们提出了一种新的思想，对切环理论中所谓的高斯周期关系进行几何推广。即利用分环多项式作为高斯周期不可约多项式产生的机制，我们引入了e个自变量y_0，【三键】y_<e-1>，并构造了e^2个有理函数u_<ij>，与定义分环数的方法类似。然后证明了Q（y_0)（三键）y_<e-1>）是Q（u'_<ij>s）的一个环扩展。通过这种方法，我们成功地构造了一个e次循环多项式的参数族；特别是对于e=7，我们发现了一个简单的族，它的系数是常数项n^7的参数n的积分多项式。这给了所谓的莱默项目一个本质上的新发展。我们注意到这个结果也给出了著名的希尔伯特第12问题的部分答案，该问题要求在给定的数域上构造阿贝尔扩展，

The main subject of our research project is the constructive sapect of the Inverse Galois theory, and our aim is to develop the systematic method to construct the family of abelian equations, which has been one of the central problems in number theory. In this research work we focused our interests to the case of cyclic equations. We proposed a new idea to make a geometric generalization of the so called Gaussian period relations in the theory of cyclotomy. Namely making use of the mechanism by which a cyclotomic polynomials give rise as irreducible polynomials of Gaussian periods, we introduced e independent variables y_0,【triple bond】y_<e-1> and constructed e^2 rational functions u_<ij> of y's, in the similar way as the cyclotomic numbers are defined. Then we proved that Q(y_0【triple bond】y_<e-1>) is a cyclic extension of Q(u'_<ij>s). By this way, we have succeeded to construct small degree e a parametric family of cyclic polynomials of degree e ; especially for e=7, we found, a simple family whose coefficients are integral polynomials in our parameter n with constant term n^7. This gives an essentially new development in the so called Lehmer project. We remark that this result gives also a partial answer to the famous 12th problem of Hilbert's, which requires to construct abelian extensions over given number field,