2004 Fiscal Year Final Research Report Summary
Distribution of units of an algebraic number field from the viewpoint of class field theory and analytic number theory
| Project/Area Number |
13640049
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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 | Meijo University |
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Principal Investigator |
KITAOKA Yoshiyuki Meijo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40022686)
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| Project Period (FY) |
2001 – 2004
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| Keywords | class field / analytic number theory / unit / distribution |
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| Research Abstract |
The aim o f this research is to study the distribution of units of an algebraic number field. There maybe several viewpoints. Ours is based on the class field theory and the analytic number theory The reason for the class field theory is the following: For an integral ideal A of an algebraic number field F, we can associate the unique abelian extension of conductor Aover F, and the extension degree is the product of the lass number of F and the residual index of residue classes represented by units in the residue class group modulo A. The class number is studied very well. But there is nothing about residual indices. As a matter of fact, almost nobody knew how to formulate the vague problem "distribution of units". So we adopted the viewpoint that the distribution of values of residual indices is nothing but the distribution of units, and we studied it using methods in the analytic number theory. At the beginning, we studied real quadratic fields and real cubic fields with negative discriminant, in detail.. The results were published in Nagoya Math. J. and J. of Number Theory. The next problem was its generalization to any algebraic number field. I completed it in the case of prime ideals. To treat more general ideals, it is necessary to study algebraic number fields in detail. For the time being, we are trying the rational prime number case and in the case that the rank of the unit group is almost over. When the rank of unit group is greater than one, the situation is much more complicated and we are collecting more information.
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