Research on Relative Class Number of an Imaginary Abelian Number Field by Means of Determinant
| Project/Area Number |
17540047
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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 | Kanazawa Institute of Technology |
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Principal Investigator |
HIRABAYASHI Mikihito Kanazawa Institute of Technology, Academic Foundations Programs, Professor, 基礎教育部, 教授 (20167612)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2006: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
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| Keywords | imaginary abelian number field / relative class number / relative class number formula / Maillet determinant / Demjanenko determinant / cyclotomic field / Newman行列式 |
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| Research Abstract |
1.Recently Endo gave a determinant formula for the quotient of the relative class numbers of a quadratic extension of a cyclotomic field K with odd conductor over that of K. (This result seems to have been unpublished.)
The head investigator has generalized the formula to an imaginary abelian number field K, by giving a formula with parameter b. If the field K is the 4th cyclotomic field and the quadratic extension is the composite of K and the 4 th cyclotomic field and if the parameter b is equal to 4p+1, then we have a formula with explicit sign, which is a refinement of Kanemitsu and Kuzumaki's. If K is a cyclotomic field with odd conductor m and the quadratic extension of K is the composite of K and the quadratic field with 2-power conductor, we have the above-mentioned formulas by taking b as 4m+1 or 8m+1. If K is the pth cyclotomic field and if b is equal to 2, we have Endo's formulas in 1996.
The investigator has presented these results and the relation among these formulas in Number Theory Seminar at Meijigakuin University and "Algebraic Number Theory and Related Topics" at Research Institute for Mathematical Sciences, Kyoto University.
2.In 1970 using a determinant, Newman gave a formula for the relative class number of the pth cyclotomic field to calculate the relative class number of the field. Skula generalized the formula to the p-power-th cyclotomic field.
The head investigator has generalizes the formulas to an imaginary abelian number field K. This generalized formula has parameter b. If K is the pth cyclotomic field and b is equal to p plus one or p-power plus one, our formula determines the sign that Newman and Skula did not assign.
The investigator has presented these results in the seminar "Algebraic Number Theory and Related Topics" at RIMS, Kyoto University.
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Report
(3 results)
Research Products
(2 results)
