2001 Fiscal Year Final Research Report Summary
p-adic analysis of algebraic numer fields and algorithm on algebraic number fields or finite fields
| Project/Area Number |
12640029
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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 | NAGASAKI UNIVERSITY |
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Principal Investigator |
KUDO Aichi Nagasaki Univ., Fac. of Engineering, Professor, 工学部, 教授 (00112285)
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| Co-Investigator(Kenkyū-buntansha) |
MARUYAMA Yukihiro Nagasaki Univ., Fac. of Economics, Professor, 経済学部, 教授 (30229629)
SUEYOSHI Yutaka Nagasaki Univ., Fac. of Education, Associate Professor, 工学部, 助教授 (80128040)
WASHIO Tadashi Nagasaki Univ., Fac. of Education, Professor, 教育学部, 教授 (60039435)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Bernoulli number / p-adic zeta-funciton / Carmichael number / elliptic curve / Hasse invariant / 4-class rank / 2-class field tower / decision process |
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| Research Abstract |
A. Kudo studied algorithms for computing p-adic properties of Bernoulli numbers, for prime factor ization of integers and for calculation of Carmichael numbers. He proved that the p-adic Euler constant γ_p of p-adic zeta function is equivalent to -(B_<p-1>/(p-1) - 1/p) for modulo p (p【greater than or equal】5), where B_n denotes the n-th Bernoulli number, and calculated the value (modp) of them for p【less than or equal】20,000,000. It is derived that in this range γ_p*0 (modp) for p≠5, 13, 563. Also he calculated all Carmichael numbers less than 10^<17> with their prime factorizations.
T. Washio studied class number and Hasse invariants of elliptic curves over finite fields. By means of determination of the number of rational points, he derived a sufficient condition for an elliptic curve over a finite prime field GF(p) to be supersingular, and gave a new equality of binomial coefficients.
Y. Sueyoshi studied 4-class ranks of quadratic number fields and matrices over finite field GF(2). Using Redei matrices, he proved new inequality relations between the narrow 4-class ranks of quadratic number fields. He further gave a characterization of Redei matrices with minimal rank. As an application of these results, he also proved the fact that if the ideal class group of imaginary quadratic number field K contains a subgroup of type (4,4, 2,2) and 4 is not contained in the prime discriminant of K, then the 2-class field tower of K is infinite.
Y. Maruyama studied discrete optimization problems using the theory of finite automata. He intro duced the notion of a new sequential decision process called bitone sequential decision process and gave a strong representation theorem for a discrete decision process.
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