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

我们研究了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次的正则循环扩展N/F有正规积分基时。由Hilbert和Speiser，理Q满足所有素数p的条件，另一方面，由Greither等人证明，对于无穷多个p，任何不同于Q的F都不满足（H_p）。因此，问哪个（F, p）满足条件就变得有趣了。正如我们在Iwasawa理论中看到的，当我们处理p扩展时，p-整数环是一个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。在这些条件下，我们得到了许多结果。最令人印象深刻的结果是它们与斯蒂克尔伯格理想之间的关系。设G为p阶有限域的乘法群，设S为群环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(ζ_p)， H=Gal（K/F）。将H作为G的子群，理想S_H可以作用于k的p-理想类群Cl_K，最重要的结果是：(1)当且仅当S_H杀死Cl_K时F满足（H_p‘），(2)我们得到了理想S_H的几个性质，并作为应用(3)我们“确定”了p-切环场满足（H_p’）的子场。

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').