Schur indices and Schur groups. II.
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 \(\chi\).
scientific article; zbMATH DE number 917731