what is $\sum_{n \in \mathbb{Z}/p \mathbb{Z}^* } \chi(n)$?

Question

Given a no trivial homomorphism $\chi$ from $\mathbb{Z}/p \mathbb{Z}^*$ to $\mathbb{C}^*$, what is $\sum_{n \in \mathbb{Z}/p \mathbb{Z}^* } \chi(n)$? is it $0$? why?

Answer 1

If $\chi:(\Bbb Z/p\Bbb Z)^\ast\to\Bbb C^\ast$ is not trivial, then there exists $m\in(\Bbb Z/p\Bbb Z)^\ast$ such that $\chi(m)\neq 1$. We have \begin{align} \chi(m)\sum_{n \in (\Bbb Z/p\Bbb Z)^\ast }\chi(n) &=\sum_{n \in (\Bbb Z/p\Bbb Z)^\ast }\chi(mn)\\ &=\sum_{n \in (\Bbb Z/p\Bbb Z)^\ast }\chi(n) \end{align} hence $$(\chi(m)-1)\sum_{n \in (\Bbb Z/p\Bbb Z)^\ast }\chi(n)$$ from which $$\sum_{n \in (\Bbb Z/p\Bbb Z)^\ast }\chi(n)=0$$