Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Canthe Galois action on all abelian (Galois) fields $K/mathbb{Q}$ be realizedexplicitly via an action on characters of some finite group? Are allsubfields of a cyclotomic field of the form $mathbb{Q}(chi)$, for someirreducible character $chi$ of a finite group G? In particular, weexplicitly determine the Galois action on all irreducible characters ofthe generalized symmetric groups. We also determine the smallestextension of $mathbb{Q}$ required to realize (using matrices) a givenirreducible representation of a generalized symmetric group.Received: 18 February 2002