2006 Fiscal Year Final Research Report Summary
Study of the structure of integer rings from the view point of cyclotomic Iwasawa theory
| Project/Area Number |
16540033
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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 | Ibaraki University (2005-2006)
Yokohama City University (2004) |
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Principal Investigator |
ICHIMURA Humio Ibaraki University, College of Science, Professor, 理学部, 教授 (00203109)
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| Co-Investigator(Kenkyū-buntansha) |
NAITO Hirotada Kagawa University, Faculty of Education, Professor, 教育学部, 教授 (00180224)
KOYA Yoshihiro Yokohama City University, International College of Arts and Science, Associate Professor, 国際総合科学部, 準教授 (50254230)
SUMIDA Hiroki Tokushima University, Graduate School of Enginnering, Associate Professor, 大学院ソシオテクノサイエンス研究部, 助教授 (90291476)
AIBA Akira Ibaraki University, College of Science, Associate Professor, 理学部, 助教授 (90202457)
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| Project Period (FY) |
2004 – 2006
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| Keywords | normal integral basis / ideal class group / Stickelberger ideal |
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| Research Abstract |
We studied on the Hilbert-Speiser number fields and Stickelberger ideals. A number field F satisfies the Hilbert-Speiser condition (H_p) at a prime p when any tame cyclic extension N/F of degree p has a normal integral basis. By Hilbert and Speiser, the rationals Q satisfy the condition for all primes p. on the other hand, it is shown by Greither et al. that any F different from Q does not satisfy (H_p) for infinitely many p. Thus, it becomes of interest to ask which (F, p) satisfies the condition. As we can see in Iwasawa theory, the ring of p-integers is an object more natural than the usual integer ring when we deal with p-extensions. We denote by (H_p') the corresponding condition for the p-integer ring. We obtained many results on these conditions. Most impreesiveresult is a relation between them and Stickelberger ideals. Let G be the multiplicative group of the finite field of order p, and let S be the classical Stickelberger ideal of the group ring Z[G]. For each subgroup H of G, we define an ideal S_H of Z[H] as a H-part of S. Let K=F(ζ_p) and H=Gal(K/F). Regarding H as a subgroup of G, the ideal S_H can act on the p-ideal class group Cl_K of K. Most important results are (1) that F satisfies (H_p') if and only if S_H kills Cl_K, (2) that we obtained several properties of the ideal S_H, and as an application (3) that we "determined" the subfields of the p-cyclotomic field satisfying (H_p').
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Research Products
(28 results)
