This is a basic question-
Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ of $\mathbb{Q}$ in $\mathbb{C}$.
Why is it true that the values of the characters of $\Gamma$, i.e. the numbers $\chi(\gamma)$ $(\chi\in R(\Gamma), \gamma\in \Gamma)$ lie in some number field $F \subset\mathbb{Q^c}$ containing $K$?
If $\Gamma$ is a finite group of order $n$ then homomorphic images of its elements must have order dividing the number $n$, in which case the eigenvalues of the corresponding linear transformations are $n$th roots of unity, so any $\Bbb Z$-linear sum of these eigenvalues (virtual characters in particular) must be contained in ${\Bbb Q}(\zeta_n)$ and hence in $F=K(\zeta_n)\supseteq K$ for any number field $K$.