If $k$ is a finite field and $\omega:k^* \to \mathbb{C}^*$ is a homomorphism s.t. $\omega^2\neq 1$, why does $\sum_{d\in k^*} \omega(d)^2 = 0$?
Came up in a computation for a problem with characters. Since $k^*$ is a cyclic group, I think $\omega(d)$ should be a root of unity. Since $\omega^2 \neq 1$, there is a generator $x$ for $k^*$ such that $\omega(x)$ is an $n$-th root of unity, $n>2$.
I am not sure where to go from here. Thanks!