(a) Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$ and state Euler's criterion.

(b) Compute $\left(\frac{-1}{p}\right)$ and prove that

$$\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$$

whenever $a$ and $b$ are coprime to $p$.

(c) Let $n$ be any integer such that $1 \leqslant n \leqslant p-2$. Let $m$ be the unique integer such that $1 \leqslant m \leqslant p-2$ and $m n \equiv 1(\bmod p)$. Prove that

$$\left(\frac{n(n+1)}{p}\right)=\left(\frac{1+m}{p}\right)$$

(d) Find

$$\sum_{n=1}^{p-2}\left(\frac{n(n+1)}{p}\right)$$