Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of n-dimensional representations of $G$ over $K^c$.
Could anyone explain the following comment made in Frohlich's book 'galois module theory of algebraic integers'--
The values of the characters of $G$, i.e. the numbers $\chi(g)$ $(\chi\in R_G, g\in G)$ lie in some number field $E$ containing $K$, which is Galois over $\mathbb{Q}$.
First of all, can $E$ be formed by adjoining the finitely many values $\chi(g)$ where $\chi$ runs over the irreducible characters in $R_G$ and $g$ runs over $G$?