State and prove Lagrange's theorem about polynomial congruences modulo a prime.

Define the Euler totient function $\phi$.

Let $p$ be a prime and let $d$ be a positive divisor of $p-1$. Show that there are exactly $\phi(d)$ elements of $(\mathbb{Z} / p \mathbb{Z})^{\times}$with order $d .$

Deduce that $(\mathbb{Z} / p \mathbb{Z})^{\times}$is cyclic.

Let $g$ be a primitive root modulo $p^{2}$. Show that $g$ must be a primitive root modulo $p$.

Let $g$ be a primitive root modulo $p$. Must it be a primitive root modulo $p^{2}$ ? Give a proof or a counterexample.