THE INVERSE PROBLEM OF GALOIS AND ITS APPLICATION TO NUMBER THEORY

1.首席调查员在1999年组织了两次讲习班，并在2000年联合举办了两次研究会议。在这些准备工作的帮助下，他在2001年的最后一年参与组织了一次名为“伽罗瓦理论和模形式”的国际会议。会议记录目前正在筹备中，准备在Kluwer Acad的DEVM系列中发表。Publ.会议的目的是收集各相关领域的信息和技术，并公布这一研究项目的结果。在该项目的三年期间，邀请了14名外国数学家，包括会议的特邀演讲者，他们对研究项目的支持是富有成效的。(1)对于奇数阶n阶循环群，他利用线性分式变换表示，在第n阶割圆域的极大实子域上构造了简单明了的单参数一般多项式族。(2)在n=3的情况下，他可以构造这样一个具有两个整参数的三次多项式族，它将所有类数可被3整除的二次域及其所有未分枝的循环三次扩张参数化；(3)在研究项目的最后一年，他还准备了一篇关于代数论和解析数论相互作用的历史论述。15名研究人员为这三年准备了73篇研究论文。其中选取10份预印本作为研究报告正文的附录二。

1. The head investigator organized two workshops in 1999, and jointly organized two research meetings in 2000. With these preparations he co-organized an International Conference "Galois Theory and Modular Forms" in the final year, 2001. The proceedings is now under preparation for publication in the DEVM Series of Kluwer Acad. Publ. The purpose of the conference is to collect informations and techniques in various related fields and to announce the results of this research project. During the three years of the project, 14 foreign mathematicians were invited, including the invited speakers of the meetings, whose supports on the research project were fruitful.2. The head investigator obtained the following results on the inverse problem of Galois : (1) for a cyclic group of odd order n he constructed simple and clear generic family of polynomials with one parameter over the maximal real sub field of the nth cylotomic field by utilizing linear fractional transformation representations. Then he generalized the results also for a dihedral group of order 2n with K. Hashimoto ; (2) in case of n=3, he could construct such a family of cubic polynomials with two integral parameters which parametrizes all of the quadratic fields with class numbers divisible by three and all of unramified cyclic cubic extensions of them ; (3) he also prepared a historical exposition on interactions between algebraic number theory and analytic number theory in the final year of the research project.3. The 15 Investigators prepared 73 research papers for the three years. Among them 10 preprints are selected and attached as Appendix II to the main body of the research report.