On the 2-class field tower of \(\mathbb Q (\sqrt{2p_1p_2},i)\) and the Galois group of its second Hilbert 2-class field
type: Item
instance of: scholarly article
title: On the 2-class field tower of \(\mathbb Q (\sqrt{2p_1p_2},i)\) and the Galois group of its second Hilbert 2-class field
zbMATH Open document ID: 1338.11099
author: Abdelmalek Azizi
author: Abdelkader Zekhnini
author: Mohammed Taous
published in: Collectanea Mathematica
publication date: 2015-01-15T00:00:00Z
review text: Given two prime numbers \(p_1\) and \(p_2\) such that \(p_1\equiv p_2 \equiv 5 \mod 8\), the authors study the 2-Hilbert class field tower of the biquadratic extension \(K=\mathbb{Q}(\sqrt{2p_1p_2}, i)\). For a field \(K\), denote by \(K'\) the 2-Hilbert class field of \(K\). The main result of the paper is that the 2-class field tower of \(K\) has length 2, and the Galois group \(G=\mathrm{Gal}((K')'/K)\) satisfies \(|G|= 2h(L) \geq 64,\) where \(L=\mathbb{Q}(\sqrt{2}, \sqrt{p_1p_2}, i).\) The authors also describe the cases where the equality holds. A complementary result is a description of the 2-part of the class group of unramified quadratic extensions of \(K\). The tools used in the proof include the class number formula, as well as reciprocity laws for the Hilbert symbol. Previous results of the authors on class numbers of biquadratic fields are also extensively used.
reviewed by: Alexey Zykin
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 6386912
zbMATH Keywords: class groups
zbMATH Keywords: class fields
zbMATH Keywords: class numbers
zbMATH Keywords: class field towers
zbMATH Keywords: Hilbert class
zbMATH Keywords: biquadratic extensions
MaRDI profile type: Publication
full work available at URL: s13348-013-0085-4
OpenAlex ID: W2134121484
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DOI: 10.1007/S13348-013-0085-4
description: scientific article; zbMATH DE number 6386912