This is a question for an assignment, so just some pointers in the right direction would be great.
Suppose that $n \in \mathbb Z$ and that $\alpha \in \mathbb Q_p$ satisfies $\alpha^2 = n$. Prove that $\alpha$ must be a $p$-adic integer.
So far I have that if $\alpha = \sum_{k=-r}^{\infty} a_kp^k$ then $$ \alpha^2 = a_{-r}p^{-2r} + 2a_{-r}a_{-r+1}p^{-2r+1} + \cdots $$ But how can I conclude that if this is an integer then $a_i = 0$ for all negative $i$?