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.

Similar facts

 * Sufficiently large implies splitting: This in particular shows that there exists at least one minimal splitting field that is contained in a cyclotomic extension of rationals.
 * Field generated by character values is splitting field implies it is the unique minimal splitting field

Opposite facts

 * Field generated by character values need not be cyclotomic
 * Minimal splitting field need not be unique
 * Splitting not implies sufficiently large
 * Field generated by character values is splitting field implies it is the unique minimal splitting field
 * Minimal splitting field need not be cyclotomic
 * Minimal splitting field need not be contained in a cyclotomic extension of rationals

Facts used

 * 1) uses::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.