2005 Fiscal Year Final Research Report Summary
Construction of Generic Polynomials in Galois Theory and application to Number Theory
| Project/Area Number |
15340015
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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 | Waseda University |
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Principal Investigator |
HASHIMOTO Kiichiro Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (90143370)
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| Co-Investigator(Kenkyū-buntansha) |
KOMATSU Keiichi Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (80092550)
MURAKAMI Jun Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (90157751)
MIYAKE Katsuya Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 客員教授 (20023632)
FUKUDA Takashi Ninon University, Faculty of Engineering, for Production, Associate Professor, 生産工学部, 助教授 (00181272)
TSUNOGAI Hiroshi Sophia University, Faculty of Science and Engineering, Assistant Professor, 理工学部, 講師 (20267412)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Galois Theory / Galois Group / Inverse Galois Problems / Generic Polynomial / Noether's Problem / Cyclic Polynomial / Meta abelian Group / Gaussian Periods |
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| Research Abstract |
Thanks to the current Grant-in-Aid, we were able to organize seven research workshops inviting the most active mathematicians on this field, through which we had many discussions on our subjects.
This enabled us to make a considerable developments along our reseach project on Galois theory.
As for the main theme of constructing generic polynomials with given finite groups over Q, our first result is the construction of concrete and simple families of quintic polynomials with two parameters for each of the five transitive permutation groups of degree 5. As a remarkable application we have established the proof of the genericity of the famous family of A_5 polynomials of degree 6 found by A.Bumer, in connection with algebraic curves of genus two whose Jacobian have real multiplication of discriminant 5.
Our second result is concerned with the Noethers' Problem for the meta abelian groups of exponent 8 which are subgroups of the affine transformation group over Z/8Z. We have proved the affirmative answer for the linear representation of degree 4 for each of them, in contrast with the negative answer for cyclic group of order 8. As a biproduct of this result, we obtained a simple criterion for a cyclic extension L/K of degree 4 to be embedded into a cyclic extension of degree 8.
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