The solvability in field of Prime Characteristic p

Question

Let $\mathbb{F}_p$ be a field of prime characteristic $p>0$.The question is what we can say about the solvability of the equation $x^2+y^2=-1$ in $\mathbb{F}_p$ for every prime $p$. I have found out that there are $\frac{p-1}{2}$ different squares in the field $\mathbb{F}_p$. I was thinking about the circle of radius $kp-1$ has integer point for some $k$.