Construction of abelian equations and study of Gaussian sums
Project/Area Number 
12640047

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Waseda University 
Principal Investigator 
HASHIMOTO Kiichiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)

CoInvestigator(Kenkyūbuntansha) 
UMEGAKI Atsuki Sophia Univ., Department of Math., Assistant, 理工学部, 助手 (60329109)
KOMATSU Keiichi Waseda Univ., Department of Math.Sci., Professor, 理工学部, 教授 (80092550)

Project Period (FY) 
2000 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)

Keywords  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 
Research Abstract 
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_<e1> 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_<e1>) 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,

Report
(4 results)
Research Products
(25 results)