Conductor of the field generated by the character values (awaiting review)

Given a complex character $\chi$ of a group $G$, define $\Q(\chi)$ to be the smallest field containing $\Q$ and the character values of $\chi$. This field is abelian and hence has a conductor, a positive integer giving the minimum $n$ so that $\Q(\chi) \subseteq \Q(\zeta_n)$.