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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?

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Yes, by Hensel's Lemma (and the generalisation also holds for the algebraic extensions of $\mathbb{Z}_p$ and $\mathbb{F}_p$). – George Lowther Apr 6 '11 at 15:45
The "and only those" part is wrong when $p=2$. – KCd Mar 5 '12 at 5:38

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.

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

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