Let $w$ be an n-th root of unity, I have two questions

1) What are the conditions on the prime $p$ such that $w\in \mathbb{Z}_p$, and if it is the case what is the p-adic expansion of an n-th root of unity in that case (do we have a closed formula of this expansion)

2) What about the other cases i.e when $w$ does not belong to $\mathbb{Z}_p$ and belongs to a finite extension of $\mathbb{Q}_p$, do we have an expression in terms of generators of this extension.


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    $\begingroup$ I think this question is more suited to math.stackexchange.com. But briefly, you can find what you need in most books about $p$-adic numbers. For example, have a look at Gouvea's "p-adic Numbers: An introduction" book. Specifically, Proposition 3.4.2. $\endgroup$ – Leray Jenkins Aug 14 at 9:55
  • $\begingroup$ Sure, what about the p-adic expansion ? $\endgroup$ – wkm Aug 14 at 10:00
  • $\begingroup$ Again, the basic ideas on how to do this is covered in the book. Just think, can you solve $x^{p-1} \equiv 1 \mod p$ and then $\mod p^2$ and so on? $\endgroup$ – Leray Jenkins Aug 14 at 10:09
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    $\begingroup$ Another approach, 143983, is to post to mathstackexchange, as has been suggested to you. But be sure to include everything you already know about the question, or else it will get closed, and be sure to include your objections to Hensel, or else you'll have to go through this all over again. $\endgroup$ – Gerry Myerson Aug 14 at 13:17
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    $\begingroup$ I've had interest in algebraic $p$-adic expansions using $p^\mathbb{Q}$ e.g. $\sqrt{-1}=1+2^{1/2}+2^{3/4}+2^{7/8}+ ... + \zeta_{3}2 + ...$ $\endgroup$ – David Lampert Aug 14 at 16:00

For $p$ odd

  • $\Bbb{Z}_p^\times = \langle\zeta_{p-1}\rangle \times (1+p)^{\Bbb{Z}_p}$ where $\zeta_{p-1} = \lim_{n \to \infty} g^{p^n}$ for $g \in \Bbb{Z}$ of order $p-1$ in $\Bbb{Z}/p\Bbb{Z}$.

  • If $K/\Bbb{Q}_p$ is a finite extension whose residue field is $O_K/(\pi) \cong \Bbb{F}_{p^f}$ then take $g\in O_K$ of order $p^f-1 $ modulo $(\pi)$ you'll have $\zeta_{p^f-1} = \lim_{n \to \infty} g^{p^{fn}}$.

  • For $p \nmid m$ let $f$ be the order of $p \bmod m$, $O_{\Bbb{Q}_p(\zeta_{p^f-1})}=O_{\Bbb{Q}_p(\zeta_m)}= \sum_{l=0}^{f-1} \zeta_m^l\Bbb{Z}_p$, it is a complete DVR with uniformizer $p$ of valuation $1$ and residue field $\Bbb{F}_{p^f}$. For the Galois actions you might prefer a normal basis for $\Bbb{F}_{p^f}/\Bbb{F}_p$.

  • $O_{\Bbb{Q}_p(\zeta_m,\zeta_{p^r})} =\sum_{m=0}^{(p-1)p^{r-1}-1}(\zeta_{p^r}-1)^m O_{\Bbb{Q}_p(\zeta_m)}$ with uniformizer $\zeta_{p^r}-1$ of valuation $(p-1)p^{r-1}$ and residue field $\Bbb{F}_{p^f}$. In particular $\Bbb{Q}_p(\zeta_{p^r})/\Bbb{Q}_p$ is totally ramified of degree $(p-1)p^{r-1}$.

Knowing a finite extension of $\Bbb{Q}_p$ means knowing its uniformizer, residue field and how the Galois group acts on both.

  • $\begingroup$ I'm looking for the general expression of coefficients! $\endgroup$ – wkm Aug 14 at 15:55
  • $\begingroup$ That's the general expression. The "coefficients" of what ? ${}{}$ $\endgroup$ – reuns Aug 14 at 16:00
  • $\begingroup$ of the p-adic expansion of a given n-th root of unity as in David Lampert comment six minutes ago $\endgroup$ – wkm Aug 14 at 16:06
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    $\begingroup$ I gave it : it is $\zeta = g^{p^n} \bmod p^n$. Do you understand that $p$-adic series and $p$-adic limit is the same ? $\endgroup$ – reuns Aug 14 at 16:08
  • $\begingroup$ Can you be more explicit please $\endgroup$ – wkm Aug 14 at 16:11

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