Arithmetic of characters of generalized symmetric groups. (Q1423768)

Theorem: If \(k/\mathbb{Q}\) is an Abelian field extension, then there exists a generalized symmetric group \(G\) and an absolutely irreducible character \(\chi\) of \(G\) such that \(k=\mathbb{Q}(\chi)\). -- Recall that \(G\) is a generalized symmetric group, if \(G\simeq S_n\wr C_l\) (wreath product of the symmetric group \(S_n\) and the cyclic group \(C_l\) of order \(l\)). Since \(S_n\wr C_l\simeq(C_l)^n\rtimes S_n\), the representation theory of such groups is well-known; moreover, by a result of \textit{M. Bénard} [J. Algebra 38, 318-342 (1976; Zbl 0327.20004)], the Schur index of each \(\chi\) is 1. This together with an explicit description of the Galois action on \(\chi\) and the Kronecker-Weber theorem are the main ingredients in the proof.