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Construction of abelian equations and study of Gaussian sums

阿贝尔方程的构造和高斯和的研究

基本信息

批准号：
12640047
负责人：
HASHIMOTO Kiichiro
金额：
$ 1.86万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640047/
关键词：
abelian equations Inverse Galois Problem Gaussian period cyclic polynomial Lehmer project Galois group period equation cycle extension 巡回方程式アーベル方程式 cyclotomic numbers ヤコビ和ガウス和 Kummer拡大

项目摘要

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,

我们的主要研究课题是逆伽罗瓦理论的构造性，目的是发展系统的方法来构造阿贝尔方程族，这一直是数论中的中心问题之一。在这项研究工作中，我们集中我们的兴趣的情况下，循环方程。提出了一种新的思想，将割圆理论中的高斯周期关系进行几何推广。即利用分圆多项式成为高斯周期的不可约多项式的机制，引入e个自变量y_0，[三键]y_<e-1>，构造了y的e^2个有理函数u_<ij>，其方法与定义分圆数的方法类似。证明了Q（y_0[三键]y_<e-1>）是Q（u ′_s）的循环扩张<ij>.通过这种方式，我们成功地构造了小次数e，即e次循环多项式的参数族;特别是对于e=7，我们发现，一个简单的族，其系数是参数n中的整多项式，其中常数项为n^7。这在所谓的Lehmer项目中给出了基本上新的发展。我们注意到，这个结果也部分地回答了著名的Hilbert第12问题，该问题要求构造给定数域上的阿贝尔扩张，