I've started reading recently on $p$-adic numbers online. Forgive me if the question is silly.
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and let $a_1, \ldots, a_k \in \mathbb{Z}_p$. If $a_1 + \cdots + a_k = n$, is it safe to conclude that at most $n$ of the summands $a_1, \ldots, a_k$, are non-zero?
I thought the lack of ordering in $\mathbb{Z}_p$ could create issues here, but probably there's no need for that? Also could someone recommend a good book to read on $p$-adics and their algebraic extensions? Thanks.