Character of Representations over C are Algebraic Integers

Theorem
Suppose $G$ is a finite group.

Let $\chi$ be the character of any $\mathbb{C}[G]$-module $(V,\rho)$.

Then $\forall g \in G, \chi (g)$ is an algebraic integer.

Proof
By the definition of character, we have


 * $\chi(g)=\operatorname{Tr}(\rho_{g})$

where $\rho \in \hom(\mathbb{C}[G],\operatorname{Aut}(V)): \vec{e}_{g} \mapsto \rho_{g}$ by definition.

Fix an arbitrary $g \in G$.

The trace of $\rho_{g}$, $\operatorname{Tr}(\rho_{g})$, is defined as the sum of the eigenvalues of $\rho_{g}$.

From Eigenvalues of G-Representation are Roots of Unity, we have that any eigenvalue $\lambda$ of $\rho_{g}$ is an $\operatorname{Ord}(g)$ root of unity (where $\operatorname{Ord}(g)$ is the order of $g$).

Since $\lambda$ satisfies the monic polynomial $x^{\operatorname{Ord}(g)}-1$, we have that $\lambda$ is an algebraic integer.

From Ring of Algebraic Integers, we have that the sum of the eigenvalues is an algebraic integer as well and thus $\chi(g)$ is an algebraic integer.