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Let $n\geq 3$, and let $p$ be a prime number $\equiv 1 $ mod $n$.

In complex numbers, we can write a primitive $n$-th root of unity as $\exp(2\pi i/n)$.

Also, by Hensel's lemma, we see that $n$-th cyclotomic field is contained in $\mathbb{Q}_p$.

Indeed, the roots of $X^n -1 = 0$ are contained in $\mathbb{Z}_p$. Denote by $\eta_n$ a primitive $n$-th root of unity in $\mathbb{Z}_p$.

Can we use the complex number $\exp(2\pi i/n)$ to figure out the expansion of $\eta_n$ in $p$-adic integers?

Sungjin Kim
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    No. First, the notation $\eta_n$ is not well-defined since there's more than one primitive $n$th root of unity in $\mathbf Z_p^\times$. Let's say you meant to ask if there is some primitive $n$th root of unity in $\mathbf Z_p^\times$ that can be computed from $\exp(2\pi i/n)$. Still the answer is no. Look at it this way: if $n = p-1$ (for $p > 3$) the question asks for a formula for a generator of the $(p-1)$th roots of unity in $\mathbf Z_p^\times$. If you could do that then reducing the formula mod $p$ gives a formula for a generator of $(\mathbf Z/p\mathbf Z)^\times$ for all $p > 3$! – KCd Oct 04 '14 at 11:48
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    That would be a constructive proof that $(\mathbf Z/p\mathbf Z)^\times$ is cyclic for all primes greater than 3, and such an approach is not known. Strictly speaking, you could work in the ring $\mathbf Z[\exp(2\pi i/(p-1)]$ and use prime ideals lying over $p$ as a "model" for $\mathbf Z/p\mathbf Z$, but that is not explicit in the way that you seek. – KCd Oct 04 '14 at 11:52
  • @KCd Can we do it better if we fix $n$ and vary $p$ within the congruence $p\equiv 1$ mod $n$? – Sungjin Kim Oct 04 '14 at 13:23
  • @KCd I understand it now. Even though we fix $n$, it still contains the case with $n=p-1$. – Sungjin Kim Oct 04 '14 at 14:22
  • Firstly, even if you would fix $n$, you would still need a general formula for a primitive $n$-th root of unity in $\mathbb{F}_p$. I don't think a simple formula for that exists, e.g. for $n=4$ it would be tantamount to factoring $p$ in $\mathbb{Z}[i]$ or writing $p$ as a sum of two squares. Second: it is not clear to me what precisely you are asking. Do you want a "generic" $p$-adic power series, say with coefficients in $\mathbb{Z}$, that upon specializing to a given prime $p$ gives a primitive $n$-th root of unity in $\mathbb{Q}_p$? But that can't exist either, again for the same reason. – R.P. Oct 04 '14 at 23:50

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