$\chi(\cdot)$ is an algebraic integer in $\mathbb{Q}$($\zeta_n$)

Question

Let $\chi(\cdot)$ be an irreducible (over $\mathbb{C}$) character of an representation in a finite Group G:

Show that $\chi(g)$ is an algebraic integer in the cyclotomic field $\mathbb{Q}$($\zeta_n$), where $\zeta_n$ := $e^{2\pi i/n}$.

Any idea? Really idealess...

Answer 1

We know that $\chi(g)=tr(M(g))$, where $g \longmapsto M(g) \in GL_n(\mathbb{C})$ is a group homomorphism. So we know that $M(g)^{|G|}=I_n$, thus $M(g)$ is similar to some diagonal matrix where diagonal coefficients are powers of $\zeta_n$ (which is an algebraic integer in said cyclotomic field). Thus, so is their sum, which is the trace of the diagonal matrix similar to $M(g)$, which is $tr(M(g))=\chi(g)$.