In this post, we discover which representations of the generalized symmetric group can be realized over a given abelian extension of .

Let be the representation defined previously, where .

Let be a subfield, where is a primitive root of unity. Assume contains the field generated by the values of the character of . Assume is Galois and let . Note if we regard as a subset of then there is an induced action of on ,

where , . This action extends to an action on .

**Key Lemma**:

In the notation above, if and only if is equivalent to under the action of on .

Let

where and .

**Theorem**: The character of has values in if and only if ,

for all and all .

This theorem is proven in this paper.

We now determine the splitting field of any irreducible character of a generalized symmetric group.

**Theorem**: Let be an irreducible character of . We have

This theorem is also proven in this paper.

In the next post we shall give an example.