Necessary condition and sufficient for certain Galois group to be metacyclic
type: Item
instance of: scholarly article
title: Necessary condition and sufficient for certain Galois group to be metacyclic
zbMATH Open document ID: 1168.11046
author: Abdelmalek Azizi
author: Mohammed Taous
published in: Annales Mathématiques Blaise Pascal
publication date: 2009-05-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10574
review text: The authors consider the biquadratic fields \(K = \mathbb Q(i,\sqrt{d}\,)\), their second Hilbert \(2\)-class fields \(K^2\), and the corresponding Galois groups \(G = \text{Gal}(K^2/K)\). They show that if \(d = 2p\) for some prime \(p \equiv 1 \bmod 4\), then the genus class field \(K^* = \mathbb Q(i,\sqrt{2},\sqrt{p}\,)\) of \(K\) has a \(2\)-class group with rank \(2\) or \(3\), and rank \(3\) if and only if \((2/p)_4 = (p/2)_4 = +1\); here \((p/q)_4\) denotes the quartic residue symbol for primes \(p, q\) satisfying \((p/q) = +1\). Next they show that the Hilbert \(2\)-class field tower of \(K\) terminates after the first step if and only if \((2/p)_4 \neq (p/2)_4\). Finally the prove that \(G\) is a nonabelian metacyclic group if and only if \((2/p)_4 = (p/2)_4 = -1\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5554661
zbMATH Keywords: Hilbert class field
zbMATH Keywords: class field tower
zbMATH Keywords: metacyclic extensions
zbMATH Keywords: Galois group
MaRDI profile type: Publication
OpenAlex ID: W2507038307
cites work: Q4518755
cites work: Sur une question de capitulation
cites work: Q5443669
cites work: Q3023573
cites work: Imaginary quadratic fields \(k\) with cyclic \(\text{Cl}_2(k^1)\)
cites work: Real quadratic fields with abelian 2-class field tower
cites work: Imaginary quadratic fields \(k\) with \(\text{Cl}_ 2(k)\simeq(2,2^ m)\) and rank \(\text{Cl}_ 2(k^ 1)=2\).
cites work: Q3241238
cites work: On a special class of \(p\)-groups
cites work: 2-groups of almost maximal class
cites work: Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94
cites work: Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen bizuadratischen Zahlkörpers
cites work: Imaginary bicyclic biquadratic fields with cyclic 2-class group
DOI: 10.5802/AMBP.255
description: scientific article; zbMATH DE number 5554661