Expression of the Relative Class Number of an Imagianry Abelian Number Field by Means of Determinant

（1）1955年Inkeri，Carlitz和Olson分别用Inkeri行列式和含Dedekind和的行列式给出了ρ阶分圆域的相对类数公式。作者将他们的结果推广到一个虚交换数域。作者提交的论文于2002财政年度被接受。此外，作者与H. Tsumura使用多个Dedekind和推广了后一个公式。论文在研究期间已提交并被接受。（2）.迄今为止，我们已经用行列式的方法得到了虚交换数域n的许多相对类数公式。行列式的矩阵由模m的整数构成，m是行列式的导体。我们也可以用整数模m^^~来构造它们，m^^~是m的倍数。作者发现由模m和模m^^~定义的两个矩阵在一定条件下是一致的。（3）1994年Girstmair利用i/p关于g的数字表达式的系数给出了具有奇素导体p的虚二次域的相对类数公式，g是模ρ的原根。本文将其推广到具有ρ-幂导体的虚阿贝尔数域，并给出了具有2-幂导体的虚阿贝尔数域的计算公式。我们的下一个问题是将上面的公式推广到一个虚阿贝尔数域。

(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.