How to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$ Let $p$ be prime and let $\mathbb{Q}_p$ denote the field of $p$-adic numbers.
Is there an elementary way to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$?
I need this result, but I cannot find a reference. Can some recommend a good book or a set of (easily available) lecture notes to help me out? 

Are you familiar about number theory? This is equivalent to saying that $x^2 \equiv -1$ in $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ iff $p \equiv 1$ (mod $4$). But this is a basic fact on number theory, since
\begin{equation*}
\Big( \frac{-1}{p} \Big) = (-1)^{\frac{p-1}{2}} = 1 \text{ iff } p \equiv 1 (\text{ mod }4)
\end{equation*}
