Show that every complex representation of a finite group $G$ is equivalent to a unitary representation. Let $\chi$ be a character of some finite group $G$ and let $g \in G$. Explain why there are roots of unity $\omega_{1}, \ldots, \omega_{d}$ such that

$$\chi\left(g^{i}\right)=\omega_{1}^{i}+\cdots+\omega_{d}^{i}$$

for all integers $i$.

For the rest of the question let $G$ be the symmetric group on some finite set. Explain why $\chi(g)=\chi\left(g^{i}\right)$ whenever $i$ is coprime to the order of $g$.

Prove that $\chi(g) \in \mathbb{Z}$.

State without proof a formula for $\sum_{g \in G} \chi(g)^{2}$ when $\chi$ is irreducible. Is there an irreducible character $\chi$ of degree at least 2 with $\chi(g) \neq 0$ for all $g \in G$ ? Explain your answer.

[You may assume basic facts about the symmetric group, and about algebraic integers, without proof. You may also use without proof the fact that $\sum_{\substack{1 \leqslant i \leqslant n \\ \operatorname{gcd}(i, n)=1}} \omega^{i} \in \mathbb{Z}$ for any $n$th root of unity $\omega .]$