Let $p$ be an odd prime number. If $n$ is an integer prime to $p$, define $\left(\frac{n}{p}\right)$.

(i) Prove that $\chi(n)=\left(\frac{n}{p}\right)$ defines a homomorphism from $(\mathbb{Z} / p \mathbb{Z})^{\times}$to the group $\{\pm 1\}$. What is the value of $\chi(-1) ?$

(ii) If $p \equiv 1 \bmod 4$, prove that $\sum_{n=1}^{p-1} \chi(n) n=0$.