Show that if $p$ is a prime number satisfying $p\equiv 1\mod 4$, $a$ is an odd positive number, and there exists $b$ such that $a^2+b^2=p$, then $a$ is a quadratic residue $\mod p$.

I know that Fermat's theorem on sums of two squares tells us that $a^2+b^2=p$ iff $p\equiv 1\mod 4$. Also I believe a form of Wilson's Theorem could be used here, although I'm not sure how.