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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