I'm reading the book "Representation Theory of finite groups" the chapter about characters. Suppose that $V$ is a KG module where $G$ is a group and $K$ is a field. Let $\chi$ be the group character. There is Theorem which states that if $K$ is a number field then $\chi(g)$ is an algebraic number for all $g \in G$.
If $T$ be the matrix representation we can easily deduce that $\{T(g)\}^n=I$ for some positive integer $n$. So all the eigenvalues are $n$th root of unity. So $\chi(g)$ is a sum of roots of unity which is algebraic.
I don't know why we need $K$ be a number field.