2003 Fiscal Year Final Research Report Summary
Expression of the Relative Class Number of an Imagianry Abelian Number Field by Means of Determinant
| Project/Area Number |
14540049
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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, Faculty of Technology, Professor, 工学部, 教授 (20167612)
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| Project Period (FY) |
2002 – 2003
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| Keywords | imaginary abelian number field / relative class number / relative class number formula / Maillet determinant / Dem'janenko determinant / Inkeri's determi-nant / Dedekind sums / Stickelberger elemenent |
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| Research Abstract |
(1).In 1955 Inkeri, Carlitz and Olson gave relative class number formulas for the ρth cyclotomic field by means of Inkeri's determinant and by a determinant with Dedekind sums, respectively. The author generalized their results to an imaginary abelian number field. The papers submitted by the author were accepted in fiscal 2002. Moreover the author with H. Tsumura generalized the latter formula using multiple Dedekind sums. The paper has been submitted and accepted during the research period.
(2).So far we have obtained many relative class number formulas for an imaginary abelian number field Κ by means of determinants. The matrices of the determinants are constructed by integers modulo m, m being the conductor of Κ. We can also construct them by integers modulo m^^~, m^^~ being a multiple of m. The author found that the two matrices defined by integers modulo m and modulo m^^~ are coincident with each other under some conditions.
(3).In 1994 Girstmair gave a relative class number formula for the imaginary quadratic field with an odd prime conductor p by using coefficients of the digit expression of i/p with respect to g, g being a primitive root of modulo ρ. The author extended it to an imaginary abelian number field with ρ-power conductor and also gave such formulas for the field with 2-power conductor. Our next problem would be to generalize the fomulas above to an imaginary abelian number field.
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Research Products
(17 results)
