2001 Fiscal Year Final Research Report Summary
THE INVERSE PROBLEM OF GALOIS AND ITS APPLICATION TO NUMBER THEORY
| Project/Area Number |
11440013
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | TOKYO METROPOLITAN UNIVERSITY |
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Principal Investigator |
MIYAKE Katsuya TOKYO METROPOLITAN UNIVERSITY, MATH., PROFESSOR, 理学研究科, 教授 (20023632)
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| Co-Investigator(Kenkyū-buntansha) |
KOIKE Masao KYUSHU UNIVERSITY, MATH. SCI, PROFESSOR, 数理学研究科, 教授 (20022733)
NAKAMURA Hiroaki TOKYO METROP. UNIV., MATH., ASSOCIATE PROFESSOR, 理学研究科, 助教授 (60217883)
NAKAMULA Ken TOKYO METROPOLITAN UNIVERSITY, MATH., PROFESSOR, 理学研究科, 教授 (80110849)
YAMAKI Hiroyoshi KUMAMOTO UNIVERSITY, MATH., PROFESSOR, 理学部, 教授 (60028199)
HASHIMOTO Ki-ichiro WASEDA UNIVERSITY, MATH., PROFESSOR, 理工学部, 教授 (90143370)
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| Project Period (FY) |
1999 – 2001
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| Keywords | INVERSE PROBLEM OF GALOIS / DIHEDRAL EXTENSIONS / GENERIC FAMILY OF POLYNOMIALS / NOETHER PROBLEM / SPACE OF MODULI / ARITHMETIC TRIANGULAR GROUPS / HYPERGEOMETRIC POLYNOMIALS / GREENBERG CONJECTURE |
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| Research Abstract |
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.
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Research Products
(15 results)
