Study of Iwasawa Theory for Cyclotomic Fields.

在1999 ~ 2000年期间，我研究了有限域上某些费马函数域的类数，固定素数分裂的实数二次域族，素数次的非ramifield Kummer扩展的积分基。在这里，我总结了主要部分（iii）的结果。对于数域F的有限扩展E/F，当对于某α∈O_E，当O_E=O_F[α]时，我们说它具有幂积分基（简称PIB）。在这里,O_E(分别地。O_F)是E（rep . f）的环整数。如果E/F是伽罗瓦，当O_E在群环O_F[Gal (E/F)]上自由时，它有一个正规积分基（简称NIB）。我得到了以下两个结果。已知素数度的未分枝Kummer扩展如果具有NIB，则具有PIB。我首先得到了这个结果的“定量”版本，然后使用Iwasawa理论的几个结果，用PIB而没有NIB构建了许多这样的例子。设p是一个奇素数，K是一个包含单位的原始p次根的虚阿贝尔域，K_∞/K是分环的z_p扩展。对于每一层K_n （K_∞/K）我用Iwasawa不变量描述了p次的K_n上的未分支Kummer扩展具有PIB的障碍。特别是，我证明了如果“格林伯格猜想”对K的最大实子域成立，那么对于足够大的n，它们有一个PIB。

During 1999〜2000, I studied (i) the class numbers of certain Fermat function fields over finite fields, (ii) the family of real quadratic fields in which a fixed prime number splits, and (iii) integral bases of unramifield Kummer extensions of prime degree. Here, I summarize the results of the main one (iii).For a finite extension E/F of a number field F, one says that it has a power integral basis (PIB for short) when O_E=O_F[α] for some α∈O_E. Here, O_E(resp.O_F) is the ring integers of E(resp.F). If E/F is Galois, it has a normal integral basis (NIB for short) when O_E is free over the group ring O_F[Gal (E/F)]. I obtained the following two results.1. It is known that an unramified Kummer extension of prime degree has a PIB if it has a NIB.I first obtained a "quantitative" version of this result, and then constructed many such examples with PIB but no NIB using several results of Iwasawa theory.2. Let p be an odd prime number, K an imaginary abelian field containing a primitive p-th root of unity, and K_∞/K the cyclotomic Z_p-extension. For each layer K_n of K_∞/K.I described the obstruction for unramified Kummer extensions over K_n of degree p to have a PIB in terms of Iwasawa invariants. In particular, I showed that they have a PIB for sufficiently large n if the "Greenberg conjecture" holds for the maximal real subfields of K.