On the $p$-adic expansion of an integer in $\mathbb{Z}_p$.

Question

I am currently reading upon $p$-adic integers, and I have a quick question related to $p$-adic expansions in $\mathbb{Z}_p$. Given a $p$-adic expansion in $\mathbb{Z}_p$, is there a criterion to determine whether the given expansion corresponds to an integer? In other words, is there a way to characterize $p$-adic expansions in $\mathbb{Z}_p$ which are images of integers?

Any help will be useful. Thanks.

Answer 1

$$a=\sum_{k\ge 0} a_k p^k,\qquad a_k\in 0\ldots p-1$$

The $p$-adic series expansion is unique.

$a\in \Bbb{Z}_{\ge 0}$ iff $a_k=0$ for $k$ large enough.

Proof: obvious

$a\in \Bbb{Z}_{< 0}$ iff $a_k=p-1$ for $k$ large enough.

Proof: note that $\sum_{k=0}^{n-1} a_k p^k+\sum_{k=n}^\infty (p-1)p^k= \sum_{k=0}^{n-1} a_k p^k-p^n$ and any negative integer $\in [-p^n,-1]$ is of this form.