Construction of abelian equations and study of Gaussian sums

我们研究的主要课题是伽罗华逆理论的构造性知识，我们的目的是发展系统的方法来构造阿贝尔微分方程组，它一直是数论中的中心问题之一。在这项研究工作中，我们将兴趣集中在循环方程的情况下。我们提出了一种新的思想，对分圆理论中的所谓的高斯周期关系进行几何推广。即利用割圆多项式作为高斯周期的不可约多项式的机理，引入了e个自变量y_0，[三键]y_&lt；e-1&gt；构造了y的e^2个有理函数u_&lt；ij&gt；，与定义割圆数的方法类似。然后证明了Q(y0[三键]y&lt；e-1&gt；)是Q(u‘&lt；ij&gt；S)的循环扩张。通过这种方法，我们成功地构造了一个小e次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，它的系数是参数n中常数项n^7的整多项式。这在所谓的Lehmer项目中有了本质上的新发展。我们指出，这一结果也部分地回答了著名的希尔伯特第12个问题，该问题要求在给定数域上构造交换扩张，

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,