# Field generated by character values is contained in a cyclotomic extension of rationals

## Statement

Suppose $G$ is a finite group and $K$ is the field generated by character values for $G$ in characteristic zero. Then, $K$ is contained in a cyclotomic extension of the rationals.

Equivalently, $K$ is a finite abelian extension of the field of rational numbers. This equivalence follows from a deep result of Galois theory called the Kronecker-Weber theorem, though the direction of relevance to us here is straightforward.

## Facts used

1. Characters are cyclotomic integers

## Proof

The proof essentially follows from Fact (1), and the observation that since the group is finite, there are only finitely many characters involved, so there is a finite cyclotomic extension containing them all.

In fact, we can use the extension taking all $d^{th}$ roots of unity where $d$ is the exponent of $G$. This is the unique smallest sufficiently large field. It is a much more nontrivial result that such a cyclotomic extension is also a splitting field, i.e., the representations themselves can be realized over such an extension. For that, see sufficiently large implies splitting.