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On bases of the Stickelberger ideal and of the group of circular units of a cyclotomic field. (English) Zbl 0744.11052

The core of the mentioned results consists in finding a basis of the Stickelberger ideal in the case of a general cyclotomic field. This was done in the case of the \(p^ n\)-th cyclotomic field (\(p\) an odd prime, \(n\) a positive integer) in the reviewer’s paper [Acta Arith. 39, 1–6 (1981; Zbl 0372.12012)] (for the minus part) and by means of this result another proof of Iwasawa’s class number formula [K. Iwasawa, Ann. Math. (2) 76, 171–179 (1962; Zbl 0125.02003)] was given. In this formula the index of the minus part of the Stickelberger ideal is given in the form of the first factor of the class number (in the case of the \(p^n\)-th cyclotomic field).
The author proves by this method Sinnott’s class number formula [W. Sinnott, Ann. Math. (2) 108, 107–134 (1978; Zbl 0395.12014)] for the first factor of a general cyclotomic field. The generalized case is essentially more difficult.

MSC:

11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
11R27 Units and factorization
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References:

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