Determination of all quaternion CM-fields with ideal class groups of exponent 2
type: Item
instance of: scholarly article
title: Determination of all quaternion CM-fields with ideal class groups of exponent 2
zbMATH Open document ID: 0952.11025
author: Stéphane Louboutin
author: Ryotaro Okazaki
published in: Osaka Journal of Mathematics
publication date: 2000-09-20T00:00:00Z
review text: In [Acta Arith. 67, 47-62 (1994; Zbl 0809.11069)], the authors of this paper showed that the class numbers of octic quaternion CM-fields \(N\) (totally complex normal octic fields whose Galois group over the rationals is the quaternion group of order \(8\)) are always even. Here they show that there are exactly two such fields whose class groups have exponent \(2\), namely \(\mathbb Q(\sqrt{-(2+\sqrt{2})(3+\sqrt{2})})\) with class number \(2\) and \(\mathbb Q(-\sqrt{5+\sqrt{5})(5+\sqrt{21})(21+2\sqrt{105})})\) with class group of type \((2,2,2)\). The proof consists of two parts: 1) an arithmetic part using class field theory showing that if the class group of \(N\) has vanishing \(4\)-rank, then the maximal real subfield \(K\) of \(N\) has odd class number, and there are at most four primes ramifying in \(K\). 2) an analytic part giving a finite list of fields \(N\) whose maximal real subfields \(K\) satisfy the conditions from 1) and whose \(2\)-class groups are annihilated by \(2\). For each of the fields from this finite list, the class group is computed, and it turns out that only the two fields above have a class group that is an elementary abelian 2-group.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 1445745
zbMATH Keywords: quaternion group
zbMATH Keywords: CM fields
zbMATH Keywords: class group
zbMATH Keywords: L-function
zbMATH Keywords: octic quaternion CM-fields
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1445745