cases when minus one is a quadratic residue

Theorem.

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

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$\left({\displaystyle \frac{-1}{p}}\right)=1$ if and only if $p\equiv 1mod4$. By Euler’s criterion

$$\left(\frac{-1}{p}\right)\equiv {(-1)}^{(p-1)/2}modp.$$

Finally, note that the integer ${\displaystyle \frac{p-1}{2}}$ is even if $p\equiv 1mod4$ and odd if $p\equiv 3mod4$.∎