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Research on Relative Class Number of an Imaginary Abelian Number Field by Means of Determinant

利用行列式研究虚阿贝尔数域的相关类数

基本信息

批准号：
17540047
负责人：
HIRABAYASHI Mikihito
金额：
$ 0.77万
依托单位：
Kanazawa Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540047/
关键词：
imaginary abelian number field relative class number relative class number formula Maillet determinant Demjanenko determinant cyclotomic field Newman行列式

项目摘要

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of a quadratic extension of a cyclotomic field K with odd conductor over that of K. (This result seems to have been unpublished.)The head investigator has generalized the formula to an imaginary abelian number field K, by giving a formula with parameter b. If the field K is the 4th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.

1.最近，Endo给出了一个关于奇导体环切场K的二次扩展的相对类数商的行列式公式（这个结果似乎还没有发表）。首席研究者通过给出一个带参数b的公式，将公式推广到虚阿贝尔数域K。如果域K是第4个环场，二次推广是K与第4个环场的复合，如果参数b等于4p+1，那么我们就得到了一个带显符号的公式，这是Kanemitsu和Kuzumaki的改进。如果K是一个奇导体m的环场，K的二次展开式是K与2功率导体二次场的复合，取b为4m+1或8m+1，就得到了上述公式。如果K是第p个环切场如果b等于2，我们有远藤1996年的公式。研究者在日本明治大学数论研讨会和日本京都大学数学科学研究所“代数数论及相关课题”上发表了这些结果和这些公式之间的关系。1970年，Newman利用行列式给出了第p个环切场的相对类数公式，用以计算场的相对类数。Skula将这个公式推广到p次切眼场。首席研究员将公式推广到虚阿贝尔数域K。这个推广公式有参数b。如果K是第p个环场，b等于p + 1或p-幂+ 1，我们的公式决定了Newman和Skula没有分配的符号。研究者在京都大学RIMS的“代数数论及相关主题”研讨会上发表了这些结果。