Q1672020
type: Item
instance of: scholarly article
title: Classification of metabelian 2-groups \(G\) with \(\mathbf{G}^{\mathrm{ab}} (\mathbf{2},\mathbf{2}^{\mathbf n})\), \(\mathbf n\geq \mathbf 2\), and rank \(\mathbf d(\mathbf G^{\prime})=\mathbf 2\). Applications to real quadratic number fields
zbMATH Open document ID: 1435.11139
author: Elliot Benjamin
author: C. Snyder
published in: Journal of Pure and Applied Algebra
publication date: 2018-09-07T00:00:00Z
review text: The authors continue their investigations whether an algebraic number field $k$ has a finite $2$-class field tower. This holds when the rank of $Cl_2(k^1)$, the $2$-part of the class group of the Hilbert class field of $k$, is at most $2$. If it equals $2$, this means that the Galois group of $k^2/k$ should be as described in the title. In Theorem 1 the authors give a complete description of such groups $G$, which depends only on (the abelianizations of) the 3 maximal subgroups and the 3 normal subgroups of index $4$ of $G$. Theorem 2 states how the rank of $Cl_2(k^1)$ depends on the different cases. Finally, these results are applied to real quadratic number fields, whose $2$-class group has rank $2$, and concludes with some numerical examples.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 6934111
zbMATH Keywords: class number
zbMATH Keywords: 2-class field tower
zbMATH Keywords: real quadratic number field
zbMATH Keywords: metabelian group
zbMATH Keywords: 2-group
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.03.004
OpenAlex ID: W2795201391
cites work: Corrigendum to: ``Some real quadratic number fields with their Hilbert 2-class field having cyclic 2-class group''.
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DOI: 10.1016/J.JPAA.2018.03.004
label: Classification of metabelian 2-groups \(G\) with \(\mathbf{G}^{\mathrm{ab}} (\mathbf{2},\mathbf{2}^{\mathbf n})\), \(\mathbf n\geq \mathbf 2\), and rank \(\mathbf d(\mathbf G^{\prime})=\mathbf 2\). Applications to real quadratic number fields
description: scientific article; zbMATH DE number 6934111