p-adic expansion of roots of unity 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.
 A: For $p$ odd 


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*$\Bbb{Z}_p^\times = \langle\zeta_{p-1}\rangle \times (1+p)^{\Bbb{Z}_p}$ 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.
