Abstract
In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized \(G^{ab}\) is of type \((2, 2^m)\), with \(m\ge 2\), to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified pro-2-extension of the cyclotomic \(\mathbb {Z}_2\)-extension of certain number fields. Illustration is given by some real quadratic fields.
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Azizi, A., Rezzougui, M. & Zekhnini, A. On the maximal unramified pro-2-extension of certain cyclotomic \(\mathbb {Z}_2\)-extensions. Period Math Hung 83, 54–66 (2021). https://doi.org/10.1007/s10998-020-00362-x
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DOI: https://doi.org/10.1007/s10998-020-00362-x
Keywords
- Iwasawa theory
- \(\mathbb {Z}_2\)-extension
- 2-Class field tower
- Real quadratic field
- 2-Class group
- Metacyclic and non-metacyclic 2-group