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

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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').