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

1.最近，Endo给出了一个关于奇导体的分圆域K的二次扩张的相对类数与K的相对类数之商的行列式公式。(This结果似乎尚未公布）。首席研究员通过给出带参数B的公式，将该公式推广到虚阿贝尔数域K。若域K是第四阶分圆域，二次扩张是K与第四阶分圆域的合成，且参数B等于4 p +1，则得到一个显号公式，它是Kanemitsu和Kuzumaki公式的一个改进.设K是具有奇导体m的分圆域，K的二次扩张是K与具有2次幂导体的二次域的合成，取B为4 m +1或8 m +1，得到上述公式。如果K是p阶分圆域，且B等于2，则在1996年我们得到了Endo公式。研究者在明治学院大学数论讨论会和京都大学数理科学研究所的“代数数论及相关问题”中提出了这些结果和这些公式之间的关系。纽曼给出了p阶分圆域的相对类数公式，用以计算域的相对类数。Skula把这个公式推广到p次分圆域上，而研究小组组长把这个公式推广到虚交换数域K上。该广义公式具有参数B。如果K是第p次分圆域和B等于p加1或p-幂加1，我们的公式确定的符号，纽曼和Skula没有分配。调查员已在研讨会上提出了这些结果“代数数论和相关主题”在RIMS，京都大学。

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of a quadratic extension of a cyclotomic field K with odd conductor over that of K. (This result seems to have been unpublished.)The head investigator has generalized the formula to an imaginary abelian number field K, by giving a formula with parameter b. If the field K is the 4th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