Theorem 1   Let $p$ be an odd prime. Then $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1 \mod 4$ .

Proof. Let $p$ be an odd prime. Notice that $p$ is congruent to either $1$ or $3$ modulo $4$ . By the definition of the Legendre symbol, we need to verify that $\displaystyle \left(\frac{-1}{p}\right) = 1$ if and only if $p\equiv 1 \mod 4$ . By Euler's criterion $$\left(\frac{-1}{p}\right)\equiv (-1)^{(p-1)/2} \mod p.$$ Finally, note that the integer $\displaystyle \frac{p-1}{2}$ is even if $p\equiv 1 \mod 4$ and odd if $p\equiv 3 \mod 4$ .