2000 Fiscal Year Final Research Report Summary
Study of Iwasawa Theory for Cyclotomic Fields.
| Project/Area Number |
11640041
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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 | Yokohama City University |
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Principal Investigator |
ICHIMURA Humio Yokohama City Univ., Faculty of Science, Prof., 理学部, 教授 (00203109)
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| Co-Investigator(Kenkyū-buntansha) |
KOYA Yoshihiro Yokohama City Univ., Faculty of Science, Assoc. Prof., 理学部, 助教授 (50254230)
NAITO Hirotada Kagawa Univ., Faculty of Education, Assoc.Prof., 教育学部, 助教授 (00180224)
NAKAJIMA Shoichi Gakushuin Univ., Faculty of Science, Professor., 理学部, 教授 (90172311)
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| Project Period (FY) |
1999 – 2000
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| Keywords | power integral basis / normal integral basis / Greenberg conjecture / 整正規基底 / 整巾基底 / 素数次不分岐Kummer拡大 |
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| Research Abstract |
During 1999〜2000, I studied (i) the class numbers of certain Fermat function fields over finite fields, (ii) the family of real quadratic fields in which a fixed prime number splits, and (iii) integral bases of unramifield Kummer extensions of prime degree. Here, I summarize the results of the main one (iii).
For a finite extension E/F of a number field F, one says that it has a power integral basis (PIB for short) when O_E=O_F[α] for some α∈O_E. Here, O_E(resp.O_F) is the ring integers of E(resp.F). If E/F is Galois, it has a normal integral basis (NIB for short) when O_E is free over the group ring O_F[Gal (E/F)]. I obtained the following two results.
1. It is known that an unramified Kummer extension of prime degree has a PIB if it has a NIB.I first obtained a "quantitative" version of this result, and then constructed many such examples with PIB but no NIB using several results of Iwasawa theory.
2. Let p be an odd prime number, K an imaginary abelian field containing a primitive p-th root of unity, and K_∞/K the cyclotomic Z_p-extension. For each layer K_n of K_∞/K.I described the obstruction for unramified Kummer extensions over K_n of degree p to have a PIB in terms of Iwasawa invariants. In particular, I showed that they have a PIB for sufficiently large n if the "Greenberg conjecture" holds for the maximal real subfields of K.
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