Schur indices and Schur groups. II.
type: Item
instance of: scholarly article
title: Schur indices and Schur groups. II.
zbMATH Open document ID: 0859.20006
author: Udo Riese
author: Peter Schmid
published in: Journal of Algebra
publication date: 1996-10-20T00:00:00Z
review text: The paper under review continues the study of Schur groups initiated by the second author in part I [ibid. 169, No. 1, 226-247 (1994; Zbl 0838.20055)]. These are ``minimal'' groups admitting characters with nontrivial Schur index, and are in connection with the Schur subgroup \(S(K)\) of the Brauer group \(B(K)\), where \(K\) is a local field of characteristic 0. The authors revisit the classification of dyadic Schur groups (which admit certain characters with nontrivial 2-local index) and give explicit generators for \(S(K)\) in terms of Schur groups. The character theoretic point of view is emphasized and Clifford theory is heavily used. Many other examples and applications are given, and the paper ends with a nice improvement of Brauer's splitting theorem. For \(\chi\in\text{Irr}(G)\), let \(n_\chi\) be the greatest common divisor of \(\text{exp}(G)\) and \(|G|/\chi(1)\). Then the cyclotomic field \(\mathbb{Q}(n_\chi)\) is a splitting field for \(\chi\).
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 917731
zbMATH Keywords: Schur groups
zbMATH Keywords: Schur index
zbMATH Keywords: Schur subgroup
zbMATH Keywords: Brauer groups
zbMATH Keywords: dyadic Schur groups
zbMATH Keywords: characters
zbMATH Keywords: explicit generators
zbMATH Keywords: Clifford theory
zbMATH Keywords: Brauer's splitting theorem
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: splitting fields
MaRDI profile type: Publication
full work available at URL: jabr.1996.0167
OpenAlex ID: W4212863065
DOI: 10.1006/JABR.1996.0167
description: scientific article; zbMATH DE number 917731