I am trying to prove that $\mathbb{Z}_p^*$ has a primitive $p - 1$-th root of unity. I already proved that $\mathbb{Z}_p^*$ has a $p - 1$-th root of unity using Hensel's lemma.
Here is my proof: if we consider the polynomial $x^{p - 1} - 1$ it is not hard to see that there exists $\alpha_1 \in \mathbb{Z}_p^*$ such that $p(\alpha_1) \equiv 0(p\mathbb{Z}_p)$. Moreover, $p'(\alpha_1) = (p - 1)\alpha_1^{p - 2} \equiv p - 1(\mod p)$. This implies that $p'(\alpha_1) \not\equiv 0(\mod p)$. In virtue of Hensel's lemma, there exists a $p - 1$-th root of unity in $\mathbb{Z}_p^*$, but I would like to have not any root but a primitive one. How can I do it?
Thank you for your help!
One way is to take the limit of $\alpha^{p^n}$ as $n$ goes to infinity where $\alpha$ is any primitive root in $\mathbb{Z}/p\mathbb{Z}$.