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

1.最近Endo给出了具有奇导体的分圆域K在K上的二次扩张的相对类数的一个行列式公式(这一结果似乎尚未发表)。首席研究者通过给出一个带参数b的公式，将该公式推广到虚阿贝尔数域K。如果域K是第四分圆域，且二次扩张是K和第四分圆域的组合，并且如果参数b等于4p+1，则我们有一个显号的公式。对Kanemitsu和Kuzumaki的公式进行了改进。如果K是具有奇导体m的分圆域，并且K的二次扩张是K和具有2次方导体的二次场的组合，则当b为4m+1或8m+1时，我们得到了上述公式。如果K是第p个分圆域，如果b等于2，则我们在1996年得到了Endo公式。研究者在京都大学数论研讨会和京都大学数学科学研究所的《代数数论及相关问题》上发表了这些结果和这些公式之间的关系。1970年，利用行列式，我们得到了上述公式Newman给出了第p个分圆域的相对类数的公式，从而计算了该域的相对类数。Skula把这个公式推广到p次方割圆域上。首席研究者把这个公式推广到虚阿贝尔数域K上。这个推广公式有参数b。如果K是第p个割圆域，且b等于p+1或p次方加1，我们的公式决定了Newman和Skula没有指定的符号。研究者在京都大学RIMS的研讨会上提出了这些结果。

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of a quadratic extension of a cyclotomic field K with odd conductor over that of K. (This result seems to have been unpublished.)The head investigator has generalized the formula to an imaginary abelian number field K, by giving a formula with parameter b. If the field K is the 4th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