2007 Fiscal Year Final Research Report Summary
Unramified Solutions of Inverse Galois Problems and their Applications
| Project/Area Number |
18540022
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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 | Kanazawa University |
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Principal Investigator |
NOMURA Akito Kanazawa University, Graduate School of Natural Science and Technology, Associate Professor (00313700)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Tatsuro Kanazawa University, Graduate School of Natural Science and Technology, Associate Professor (90015909)
HIRABAYASHI Mikihito Kanazawa Institute of Technology, Academic Foundations Programs, Professor (20167612)
KIMURA Iwao University of Toyama, Graduate School of Science and Engineeringfor Research, Associate Professor (10313587)
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| Project Period (FY) |
2006 – 2007
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| Keywords | inverse Galois problem / embedding problem / unramified extension / ramification / class field tower |
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| Research Abstract |
Head investigator Nomura studied the existence of unramified 3-extensions over cyclic cubic fields and gave a sufficient condition that the length of the 3-class field tower of a cyclic cubic field is greater than 1.
Nomura also studied the number of primes which are ramified in G-extension. Let p be an odd prime number. Scholz and Reichardt proved that every p-group G can be realized as the Galois group of some extension M of the rational number field Q. For a finite p-group G, let t-ram(G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at t-ram(G) finite primes. We denote by d(G) the minimal number of generators of a finite p-group G. Then it is well-known d(G) t-ram(G) ≦n, where p^n is the order of G. Plans(2004)proved a better upper bound for t-ram(G). Nomura proved an improvement of the result of Plans. Nomura also proved that t-ram(G) =d(G) for any 3-group G of order less than or equal to 243.
Investigator Ito cooperated in this research by group theoretical consideration. And considerations by Hirabayashi and Kimura concerning the ideal class group played an important role to this research.
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Research Products
(37 results)
