2005 Fiscal Year Final Research Report Summary
Unramified Solutions of Inverse Galois Problems and their Applications to the Class Field Tower Problems
| Project/Area Number |
16540017
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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, 自然科学研究科, 助教授 (00313700)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Tatsuro Kanazawa University, Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (90015909)
HIRABAYASHI Mikihito Kanazawa Institute of Technology, Academic Foundations Programs, Professor, 基礎教育部, 教授 (20167612)
FUJISAKI Hiroshi Kanazawa University, Graduate School of Natural Science and Technology, lecturer, 自然科学研究科, 講師 (80304757)
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| Project Period (FY) |
2004 – 2005
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| Keywords | embedding problem / unramified extension / class number / inverse Galois problem |
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| Research Abstract |
Head investigator Nomura studied the existence of unramified 3-extensions over cyclic cubic fields. In particular we treated the following problem.
Problem P(F,G) : For a given Galois extension F/Q and a finite group G, does there exists an unramified Galois extension M/F such that the Galois group Gal(M/F) is isomorphic to G.
Let F and K be the cyclic cubic fields satisfying the certain ramification conditions. Let G_1 and G_2 be the non-abelian 3-group such that the GAP-number is [81,9] and [243,2] respectively. One of main results is stated as follows.
Assume that the class number of F is divisible by 81. Then, (1) there exists an unramified extension M/K such that the Galois group Gal(M/K) is isomorphic to G_1, (2) there exists an unramified extension M/F such that the Galois group Gal(M/F) is isomorphic to G_2.
We also studied the class number relation between certain cubic fields, and gave an alternative proof of the Naito's result.
Investigator Hirabayashi constructed some multiple Dedekind sums and gave a relative class number formula for an imaginary abelian number field by means of such Dedekind sums. He also gave a generalization of Girstmair's formula to an imaginary abelian number field.
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Research Products
(31 results)
