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The determination of the imaginary abelian number fields with class number one. (English) Zbl 0798.11046

Extending earlier work of K. Uchida [Tôhoku Math. J., II. Ser. 24, 487-499 (1972; Zbl 0248.12007), Proc. Int. Conf. Katala/Japan 1986, 151-170 (1986; Zbl 0612.12011)] and J. Masley and H. Montgomery [J. Reine Angew. Math. 286/287, 248-256 (1976; Zbl 0335.12013)] the author determines all imaginary abelian number fields with class number one. This requires the discussion of several cases, the details of which are rather technical. There are 172 such fields of all even degrees \(n \leq 24\) except for \(n=22\). The paper also contains interesting observations on fields which are “unramified-closed”.
Reviewer: M.Pohst (Berlin)

MSC:

11R20 Other abelian and metabelian extensions
11Y40 Algebraic number theory computations
11R29 Class numbers, class groups, discriminants
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