What do the $p$-adic roots of unity look like?

Question

I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, restricted to just the roots of unity, bijective?

Answer 1

This follows from the fact that $x^{p-1} - 1$ is relatively prime to its formal derivative over $\mathbb{F}_p$, which is $-x^{p-2}$.

Answer 2

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.