What do the $p$-adic roots of unity look like?
Solution 1:
Indeed, the $(p-1)^{st}$ roots of unity are the so-called Teichmüller lifts of the non-zero elements of $\mathbb{F}_p$. This construction is very important, because it generalises to Witt vectors, as the article explains, and those are widely used in number theory.
Solution 2:
This follows from the fact that $x^{p-1} - 1$ is relatively prime to its formal derivative over $\mathbb{F}_p$, which is $-x^{p-2}$.