Steinitz classes of relative Galois extensions of 2-power degree and embedding problems
type: Item
instance of: scholarly article
title: Steinitz classes of relative Galois extensions of 2-power degree and embedding problems
zbMATH Open document ID: 0922.11094
author: Bouchaïb Sodaïgui
published in: Illinois Journal of Mathematics
publication date: 1999-07-11T00:00:00Z
review text: Let \(K/k\) be a normal extension of number fields of degree \(n\), and let \({\mathcal O}_K\) denote the ring of integers of \(N\). As an \({\mathcal O}_k\)-module, it is isomorphic to \({\mathcal O}_k^{n-1} \oplus I\) for some ideal \(I\) in \({\mathcal O}_k\), whose ideal class is called the Steinitz class of the extension \(K/k\). The author continues recent investigations of the question of which ideal classes of \(k\) can be realized as Steinitz classes as \(N\) runs through certain classes of tamely ramified extensions of \(k\) with given Galois group. His main results are the following: every ideal class \(c\) of \(k\) can be realized as the Steinitz class of one of the following extensions: 1) a quadratic extension of \(k\) that can be embedded into a tamely ramified cyclic quartic extension of \(k\); 2) a biquadratic extension of \(k\) that can be embedded into a tamely ramified quaternion extension of degree \(8\) over \(k\); 3) a tamely ramified cyclic quartic extension; 4) a tamely ramified quaternion extension of \(k\) of degree \(8\). For 2) he has to assume that \(k\) contains a fourth root of unity \(i\) or that \(k(i)/k\) is ramified, while for 3) and 4) he assumes that the class number of \(k\) is odd. These results are proved by realizing these extensions more or less explicitly as ray class fields.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 1274758
zbMATH Keywords: Steinitz classes
zbMATH Keywords: quaternion extensions
zbMATH Keywords: ray class fields
zbMATH Keywords: embeddings
zbMATH Keywords: ideal class
zbMATH Keywords: tamely ramified extensions
zbMATH Keywords: Galois group
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1274758