2002 Fiscal Year Final Research Report Summary
Construction of abelian equations and study of Gaussian sums
| Project/Area Number |
12640047
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Waseda University |
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Principal Investigator |
HASHIMOTO Kiichiro Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90143370)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
2000 – 2002
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| Keywords | abelian equations / Inverse Galois Problem / Gaussian period / cyclic polynomial / Lehmer project / Galois group / period equation / cycle extension |
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| 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_<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,
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