Let $G$ be the units of $\mathbb Z/ q \mathbb Z.$ Let $\chi$ denote Dirichlet character on $G.$ I have two questions.
Let $a \in G$ with order $k, $Then it is easy to see that $\chi(a)$ is a $k$ th root of unity. I have to show that for each $k$ th root of unity occurs precisely $ϕ(q)/k$ times among the numbers $χ(a)$ as $χ$ runs over the $ϕ(q)$ Dirichlet characters. My idea is to take a homomorphism $\hat G \to \{k \text{ th roots unity} \}$ defined by $\chi \mapsto \chi(a)/\zeta$ for a fixed $\zeta.$ The kernel is $\{\chi : \chi(a) = \zeta \},$ so it is sufficient to show that the map is surjective which I am not getting how to do.
Suppose $\chi$ has order $ k, $ then again $\chi(a)$ is $k$ th root of unity. I need to show that $k$ th root of unity occurs precisely $ϕ(q)/k$ times among the numbers $χ(a)$ as $a$ runs over $G.$ In this case my attempt is to define the following homomorphism $G \to \{k \text{ th roots unity} \}$ defined by $a \mapsto \chi(a)/\zeta. $ The kernel is $ \{a : \chi(a) = \zeta \} $ so again it is enough to show the map is onto in which I need aid.
Are there are any other methods to do this?