| Project/Area Number |
15540025
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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 | Kyoto Institute of Technology |
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Principal Investigator |
ASADA Mamoru Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (30192462)
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| Co-Investigator(Kenkyū-buntansha) |
MIKI Hiroo Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (90107368)
MAITANI Fumio Kyoto Institute of Technology, Faculty of Engineering and Design, professor, 工芸学部, 教授 (10029340)
YAGASAKI Tatsuhiko Kyoto Institute of Technology, Faculty of Engineering and Design, associate professor, 工芸学部, 助教授 (40191077)
NAKAMURA Hiroaki Okayama University, Faculty of Science, professor, 理学部, 教授 (60217883)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | Galois group / cyclotomic field / Iwasawa Theory |
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| Research Abstract |
Let κ_0 be a finite algebraic number field, κ_∞ be the field obtained by adjoining to κ_0 all roots of unity, and L be the maximal unramified abelian extension of κ_∞. Let κ_1 be the field obtained by adjoining ζ_4 and ζ_p for all odd prime p to κ_0 and consider the subgroup g=Gal(κ_∞/κ_1) of Gal(κ_∞/κ_0). In this research, we have investigated the structure of Gal(L/κ_∞) and the ideal class group C_∞ of κ_∞ with this g-action.
As for Gal(L/κ_∞), we have shown that it is, as modules over the completed group algebra Z^^^[[g]], isomorphic to the direct pruduct of countable number of copies of Z^^^[[g]]. (Z^^^: the profinite completion of the ring of rational integers Z.)
On the other hand, the ideal class group C_∞ is a discrete g-module. Assume that κ_0 is totally real and p is an odd prime. Let C_∞ (p)^- denote the minus part of C_∞)(p) under the action of the complex conjugation. (A result of Kurihara indicates that the plus part is {0}.) In general, for a pro-p g-module X and the group W(p) of all p-powerth roots of unity, let Hom(X,W(p)) denote the set of continuous homomorphisms from X to W(p). Then this is naturally a discrete g-module. As for C_∞(p)^-, we have shown that it is isomorphic to Hom(Π^∞_<N=1> Z_p[[g]], W(p)). (Z_p[[g]] : the completed group algebra g over the ring of p-adic integers Z_p.
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