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Study of the structure of integer rings from the view point of cyclotomic Iwasawa theory

从分圆岩泽理论角度研究整数环的结构

基本信息

批准号：
16540033
负责人：
ICHIMURA Humio
金额：
$ 1.47万
依托单位：
Ibaraki University (2005-2006)Yokohama City University (2004)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540033/
关键词：
normal integral basis ideal class group Stickelberger ideal p-整数環単項化定理 descent Kummer拡大 Kummer duality

项目摘要

We studied on the Hilbert-Speiser number fields and Stickelberger ideals. A number field F satisfies the Hilbert-Speiser condition (H_p) at a prime p when any tame cyclic extension N/F of degree p has a normal integral basis. By Hilbert and Speiser, the rationals Q satisfy the condition for all primes p. on the other hand, it is shown by Greither et al. that any F different from Q does not satisfy (H_p) for infinitely many p. Thus, it becomes of interest to ask which (F, p) satisfies the condition. As we can see in Iwasawa theory, the ring of p-integers is an object more natural than the usual integer ring when we deal with p-extensions. We denote by (H_p') the corresponding condition for the p-integer ring. We obtained many results on these conditions. Most impreesiveresult is a relation between them and Stickelberger ideals. Let G be the multiplicative group of the finite field of order p, and let S be the classical Stickelberger ideal of the group ring Z[G]. For each subgroup H of G, we define an ideal S_H of Z[H] as a H-part of S. Let K=F(ζ_p) and H=Gal(K/F). Regarding H as a subgroup of G, the ideal S_H can act on the p-ideal class group Cl_K of K. Most important results are (1) that F satisfies (H_p') if and only if S_H kills Cl_K, (2) that we obtained several properties of the ideal S_H, and as an application (3) that we "determined" the subfields of the p-cyclotomic field satisfying (H_p').

研究了Hilbert-Speiser数域和Stickelberger理想。数域F在素数p处满足Hilbert-Speiser条件（H_p），当任意p次的驯服循环扩张N/F有正规整基时. Hilbert和Speiser证明了有理数Q对所有素数p都满足条件，而Greither等人证明了，对于无穷多个p，任何不同于Q的F都不满足（H_p）。正如我们在岩泽理论中所看到的，当我们处理p-扩张时，p-整数环是比通常的整数环更自然的对象。我们用（H_p '）表示p-整数环的相应条件。我们在这些条件下得到了许多结果。最重要的结果是它们与Stickelberger理想之间的关系。设G是p阶有限域的乘法群，S是群环Z[G]的经典Stickelberger理想.对于G的每个子群H，我们定义Z[H]的理想S_H为S的H-部分。设K=F（K_p），H=Gal（K/F）.设H是G的子群，理想S_H可以作用在K的p-理想类群Cl_K上。最重要的结果是（1）F满足（H_p '）当且仅当S_H消灭Cl_K，（2）我们得到了理想S_H的几个性质，作为应用（3）我们“确定”了满足（H_p'）的p-分圆域的子域。