2001 Fiscal Year Final Research Report Summary
Resarch on ideal class groups and the distribution of primes
| Project/Area Number |
11640046
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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 | Tokyo University of Science |
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Principal Investigator |
TAKASHI Agoh Tokyo University of Science,fac of Tech.and sci,professor, 理工学部, 教授 (60112893)
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| Co-Investigator(Kenkyū-buntansha) |
TAMIO Hara Tokyo University of Science,fac of Tech.,Ass.professor, 工学部, 助教授 (10120205)
TAKAO Kobayashi Tokyo University of Science,fac of Tech.and sci,Ass.professor, 理工学部, 助教授 (90178319)
TOSHIAKI Shoji Tokyo University of Science,fac of Tech.and sci,professor, 理工学部, 教授 (40120191)
RYUICHI Tanaka Tokyo University of Science,fac of Tech.and sci,Lecture, 理工学部, 講師 (10112898)
TOSHIKO Hosoh Tokyo University of Science,fac of Tech.and sci,Lecture, 理工学部, 講師 (30130339)
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| Project Period (FY) |
1999 – 2001
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| Keywords | ideal class group / Stickelberger ideal / Class number and unit / Bermoulli number / regular・irregular prime / zeta function / distribution of primes / sieve method |
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| Research Abstract |
On ideal calss groups we first calculated class numbers of cyclotomic fields and group indices of Stickelberger ideals in a group ring using computer, and deduced some formulae for relative class numbers by observing specific properties of Stckelberger subideals related to the Kummer-Mirimanoff sysytem of congruences. Consequently, a certain partial structure of ideal class group could be elucidated. Further, we investigated the Ankeny-Artin-Chowla conjecture and some applicable necessary and sufficient conditions were obtained. On the distribution of primes we treated some secial primes (e. g. irregular primes, SG primes, twin primes, primes of the form X^2+1, Wilson primes, and others) and derived the Legendre type prime counting functions for SG primes and primes of the form X^2+1.Using these it was pos sible to estimate upper bounds for the number of these primes. The p-divisibility problem of the class number of the p-th cyclotomic field is deeply concerned with a behavior of Bernoulli numbers. And then, we mainly studied generalizations of Lehmer's congruences, recurrences of special types and Voronoi'type congruences to obtain definite results. On the other hand, we could exploit a new relation involvi ing Bernoulli numbers and Fermat-Euler quotients, which led without difficulty to several important properties of class numbers of quadratic fields.
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