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Splitting fields of finite groups. (English. Russian original) Zbl 1270.20005
Izv. Math. 76, No. 6, 1163-1174 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 6, 95-106 (2012).
The author gives a proof of a theorem of D. M. Goldschmidt and I. M. Isaacs [J. Algebra 33, 191-199 (1975; Zbl 0297.20018)] on the Schur index of an irreducible character of a finite group when a certain Galois group is a cyclic \(p\)-group (\(p\) some prime). Using this proof, the author generalizes this theorem in the case when \(p=2\).
He also obtains some related results such as the following. If \(\chi\) is an irreducible character of a finite group \(G\) of exponent \(n\) with \(n\neq 4\) and \(m\) is the order of a maximal cyclic subgroup of \((\mathbb Z/n\mathbb Z)^*\) then the rational Schur index of \(\chi\) divides \(\varphi(n)/m\) (where \(\varphi\) is the Euler function).

MSC:
20C15 Ordinary representations and characters
12F10 Separable extensions, Galois theory
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