(i) Prove Wilson's theorem: if $p$ is prime then $(p-1) ! \equiv-1(\bmod p)$.

Deduce that if $p \equiv 1(\bmod 4)$ then

$$\left(\left(\frac{p-1}{2}\right) !\right)^{2} \equiv-1 \quad(\bmod p)$$

(ii) Suppose that $p$ is a prime of the form $4 k+3$. Show that if $x^{4} \equiv 1(\bmod p)$ then $x^{2} \equiv 1(\bmod p)$.

(iii) Deduce that if $p$ is an odd prime, then the congruence

$$x^{2} \equiv-1 \quad(\bmod p)$$

has exactly two solutions ( $\operatorname{modulo} p)$ if $p \equiv 1(\bmod 4)$, and none otherwise.