Let $p>2$ be a prime.

(a) What does it mean to say that an integer $g$ is a primitive root $\bmod p$ ?

(b) Let $k$ be an integer with $0 \leqslant k<p-1$. Let

$$S_{k}=\sum_{x=0}^{p-1} x^{k}$$

Show that $S_{k} \equiv 0(\bmod p)$. [Recall that by convention $0^{0}=1$.]

(c) Let $f(X, Y, Z)=a X^{2}+b Y^{2}+c Z^{2}$ for some $a, b, c \in \mathbb{Z}$, and let $g=1-f^{p-1}$. Show that for any $x, y, z \in \mathbb{Z}, g(x, y, z) \equiv 0$ or $1(\bmod p)$, and that

$$\sum_{x, y, z \in\{0,1, \ldots, p-1\}} g(x, y, z) \equiv 0 \quad(\bmod p) .$$

Hence show that there exist integers $x, y, z$, not all divisible by $p$, such that $f(x, y, z) \equiv 0$ $(\bmod p)$.