Thus, the Legendre Symbol (I presume you learned them by now),
Since -b^2 is a quadradic residue of a^2 we have,
That is a famous case and only happens when,
p≡1 (mod 4)
This looks greatly similar to Fermat's theorem about expressing 4k+1 primes as a sum of two unique squares.
The proof of that is much more complicated (I wonder how Fermat proved it )