On the units of unramified cyclic cubic extensions of some subfields of \(\mathbb Q\left(\sqrt d,\sqrt{-3}\right)\)
type: Item
instance of: scholarly article
title: On the units of unramified cyclic cubic extensions of some subfields of \(\mathbb Q\left(\sqrt d,\sqrt{-3}\right)\)
zbMATH Open document ID: 1187.11040
author: Abdelmalek Azizi
author: Moulay Chrif Ismaili
author: Mohamed Talbi
author: Mohamed A. Ayadi
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/10573
review text: Let \(d\) be a positive square-free integer. This article concerns the global units in cubic Galois extensions of the field \(k\), where \(k\) is either \(\mathbb Q(\sqrt d)\) or \(\mathbb Q(\sqrt d,\sqrt{-3})\). To be more precise, let us first consider the case where \(k=\mathbb Q(\sqrt d)\). We take \(K\) to be a cubic Galois extension of \(k\) which is unramified over \(k\) and dihedral over \(\mathbb Q\). According to [\textit{N. Moser}, Unités et nombre de classes d'une extension diédrale de \(\mathbb Q\), Journées Arithmétiques de Bordeaux, Astérisque 24--25, 29--35 (1975; Zbl 0313.12002)], the structure of the group of units in \(K\) can take one of five forms determined by certain unit and norm indices. The present article denotes these five structures \(\alpha,\beta,\gamma,\delta\) and \(\varepsilon\). The main result, Théorème 2.6, says that if the 3-class group of \(K\) is isomorphic to \(\mathbb Z/3\mathbb Z\times\mathbb Z/3\mathbb Z\), then in fact only two of the five structures are possible, those denoted \(\alpha\) and \(\delta\). It further states that, under these assumptions, \(\delta\) is equivalent to each of the (equivalent) statements (i) The extension \(K/k\) admits a non-trivial strongly ambiguous ideal class. (ii) \(K\) admits a Minkowski unit. (We remark that there appears to be an error in the statement of Théorème 2.6, since it claims that only the structures \(\gamma\) and \(\delta\) are possible, whereas reading the rest of the paper and the proof of that theorem, it is clear that \(\gamma\) should be replaced by \(\alpha\).) A number of computer-generated examples are given where \(K\) and \(k\) are made explicit and the isomorphism type \((\alpha\) or \(\delta)\) of the units in \(K\) determined. In particular, this allows one to read off fields \(K\) for which there exists a Minkowski unit. Attending to the case where \(k=\mathbb Q(\sqrt d,\sqrt{-3})\), the article again takes a cubic Galois extension of \(k\) which is unramified o…
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5554660
zbMATH Keywords: quartic and biquadratic fields
zbMATH Keywords: unit
zbMATH Keywords: 3-number class
reviewed by: Paul Buckingham
MaRDI profile type: Publication
OpenAlex ID: W2296369589
cites work: Units of certain imaginary abelian number fields over \(\mathbb Q\).
cites work: Ranks of 3-class groups of non-Galois cubic fields
cites work: Q4063245
cites work: Parametrization of the quadratic fields whose class numbers are divisible by three
cites work: Class groups of dihedral extensions
cites work: Q4071840
cites work: On the 3-class field tower of some biquadratic fields
DOI: 10.5802/AMBP.254
description: scientific article; zbMATH DE number 5554660