Abstract.
The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the “smallest” extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group.
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Received: 18 February 2002
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Joyner, D. Arithmetic of characters of generalized symmetric groups. Arch. Math. 81, 113–120 (2003). https://doi.org/10.1007/s00013-003-0814-0
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DOI: https://doi.org/10.1007/s00013-003-0814-0