(i) Let $p$ be a prime number. Prove that the multiplicative group of the field with $p$ elements is cyclic.

(ii) Let $p$ be an odd prime, and let $k \geqslant 1$ be an integer. Prove that we have $x^{2} \equiv 1 \bmod p^{k}$ if and only if either $x \equiv 1 \bmod p^{k}$ or $x \equiv-1 \bmod p^{k}$. Is this statement true when $p=2$ ?

Let $m$ be an odd positive integer, and let $r$ be the number of distinct prime factors of $m$. Prove that there are precisely $2^{r}$ different integers $x$ satisfying $x^{2} \equiv 1 \bmod m$ and $0<x<m$.