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Zbl 0687.12003
Gold, Robert; Kim, Jaemoon
Bases for cyclotomic units. (English)
[J] Compos. Math. 71, No.1, 13-27 (1989). ISSN 0010-437X; ISSN 1570-5846/e

Let $U\sb n=E\sb n\cap V\sb n$, the group of cyclotomic units of the field $k={\bbfQ}(\zeta\sb n)$, $\zeta\sb n=\exp (2\pi i/n)$, $n\in {\bbfN}$, $n\not\equiv 2$ (mod 4). Here $E\sb n$ is the group of units of k and $V\sb n$ is the subgroup of $k\sp{\times}$ generated by $\{\pm \zeta\sb n,1-\zeta\sp a\sb n\vert$ $1\le a<n\}$. A main goal of the article is to provide a basis for $U\sb n$, and to use this basis to show that $U\sp G\sb n=U\sb m$ for all m $\vert n$, where $G=Gal({\bbfQ}(\zeta\sb n)/{\bbfQ}(\zeta\sb m))$. There are some applications (for example, to Greenberg's conjecture concerning the p- primary part of $E\sb n/U\sb n).$ \par \{Reviewer's remark: In connection with this paper see also {\it R. Kučera} [A basis of Stickelberger's ideal and a system of main cyclotomic units of a cyclotomic field. Zap. Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklova 175, 69-74 (1989)].\}
[I.Sh.Slavutskij]

MSC 2000:: *11R27 Units and factorization
11R18 Cyclotomic extensions

Keywords: group of cyclotomic units; basis; Greenberg's conjecture

Cited in: Zbl 0972.11110 Zbl 0869.11082 Zbl 0776.11066

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