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Initial stage of a \(\mathbb{Z}_ \ell\)-extension. (Etage initial d’une \(\mathbb{Z}_ \ell\)-extension.) (French) Zbl 0799.11047

Let \(F\) be a number field and \(l\) a prime number. The paper studies \(M/F\) the maximal abelian extension unramified outside \(l\); it contains \(Z/F\) the compositum of the \(\mathbb{Z}_ l\) extensions of \(F\) as well as \(H/F\) the Bertrandias-Payan extension (= union of the cyclic extensions locally embeddable in a \(\mathbb{Z}_ l\)-extension). The maximal extensions of exponent \(l\) are \(M'\), \(Z'\) and \(H'\). They correspond to the Kummer radicals \(M\), \(Z\) and \(H\), provided that \(F\) contains the \(l\) roots of 1.
The topic of the paper is the computation of these groups in the case where \(F= \mathbb{Q}(\zeta_ 3, \sqrt{d})\) for around 700 values of \(d\). The computation uses the PARI software designed in Bordeaux.

MSC:

11R20 Other abelian and metabelian extensions
11R23 Iwasawa theory

Software:

PARI/GP
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Full Text: DOI EuDML

References:

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