Research on class numbers of cyclotomic fields and properties of Bernoulli numbers

(1)为了阐明理想类群的结构，我们研究了奇素数p的p次环切场的相对类数h^-_p。我们首先构造了对应于Maillet和Inkeri型矩阵（其行列式表示h^-_p）的Stickelberger理想的特殊基。然后，利用系数与Inkeri矩阵项相关的多项式与一个特殊的分环多项式的结果，得到了Lehmer型公式。由此，我们可以找到h^-_p的新因子，并推导出h^-_p的每个素数因子必须满足的一定条件。(2)针对与h^-_p有关的伯努利数的特殊性质，提出了含费马-欧拉商的Voronoi型同余。这在各个方面都有很多应用。例如，构造p进l函数所需的Kummer型公式和著名的Vandiver同余可以由这个同余得到。(3)用组合方法研究了一类和二类斯特林数的若干卷积，推导出了一类高阶伯努利数和二类伯努利数的各种新的性质和递推式。我们的结果几乎涵盖了所有已知的结果，因此我们相信这些结果将对伯努利数的进一步研究做出不小的贡献。

(1)To elucidate the structure of ideal class group, we studied the relative class number h^-_p of the p-th cyclotomic field for an odd prime p. We first constructed special bases of the Stickelberger ideal corresponding to the Maillet and Inkeri type matrices whose determinants represent h^-_p. Next, we obtained a Lehmer type formula by making use of the resultant of the polynomial with coefficients related to entries of Inkeri's matrix and a special cyclotomic polynomial. Consequently, we could find new factors of h^-_p and deduced a certain condition for which each prime factor of h^-_p must satisfy. (2)Concerning special properties on Bernoulli numbers connected with h^-_p, we devised the Voronoi type congruence involving the Fermat-Euler quotient. This has many applications in various aspects. For example, Kummer type formula which is needed for the construction of the p-adic L-function and the famous Vandiver congruence can be given from this congruence. (3)We studied some convolutions of the Stirling numbers of the first and second kinds by combinatorial methods and deduced various new properties and recurrences for Bernoulli numbers of the higher order and of the second kind. Our results cover almost of all the known results, and so we believe that these will contribute in no small way to a further research on Bernoulli numbers.