Study of Iwasawa Theory for Cyclotomic Fields.

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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.