1991 Fiscal Year Final Research Report Summary
Congruence relations between class numbers of cyclotomic fields
| Project/Area Number |
02640063
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Saga University |
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Principal Investigator |
UEHARA Tsuyoshi Saga Univ. Fac. of Science and Engrg, Professor, 理工学部, 教授 (80093970)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAI Shigeo Saga Univ., Fac. of Education, Ass. Prof., 教育学部, 助教授 (30186043)
KOZAKI Masanori Saga Univ., Fac. of Liberal Arts, Professor, 教養部, 教授 (70039262)
SUGITA Hiroshi Saga Univ., Fac. of Science and Engrg, Ass. Prof., 理工学部, 助教授 (50192125)
ICHIKAWA Takashi Saga Univ., Fac. of Science and Engrg, Instructor, 理工学部, 講師 (20201923)
NAKAHARA Toru Saga Univ., Fac. of Science and Engrg, Professor, 理工学部, 教授 (50039278)
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| Project Period (FY) |
1990 – 1991
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| Keywords | Cyclotomic Field / Class Number / Congruence Relation / Quadratic Field / Ideal Class Group / Unit Group / Factorizing Algorithm / Elliptic Curve |
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| Research Abstract |
In this project, we have studied mainly about congruence relations for class numbers and units of cyclotomic fields, in particular of quadratic fields.
We firstly established (cf. [1]) a general linear congruence relation modulo an odd prime number between the class numbers and units of the quadratic fields whose discriminants are divisible by 8m, where m>O is an odd square-free rational integer. From this relation we deduced various congruences for class numbers and units of two quadratic fields and indicated how they included known results by many authors. Using Stickelberger's relation we secondly gave (cf. [2)) a congruence modulo 8 between Jacobi sums and the class number of a real quadratic field, and as an application of this congruence we proved a new type of congruence for the class numbers of a real quadratic field and the corresponding imaginary one. In the proof we developed a new method of calculation of 2-adic logarithm of Jacobi sums. We believe that this method is fundamental to obtain a general congruence modulo some power of a prime number between class numbers of cyclotomic fields. Thirdly we were concerned with algorithm of factorization of large natural numbers into prime numbers, and studied a sieve method using polynomials of degree 3. Finally, during our studies of a factorizing algorithm by elliptic curves, we found two simple proof of the points of an elliptic curve over any field forming an abelian group ; one use only elementary calculus, and the other is done by computer.
Each investigator obtained good result for a respective research plan.
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