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?
What do the $p$-adic roots of unity look like?
10
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number-theory
p-adic-number-theory
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5Yes, by Hensel's Lemma (and the generalisation also holds for the algebraic extensions of $\mathbb{Z}_p$ and $\mathbb{F}_p$). http://en.wikipedia.org/wiki/Hensel%27s_lemma – 2011-04-06
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2The "and only those" part is wrong when $p=2$. – 2012-03-05