cases when minus one is a quadratic residue1IsQuadraticResidueIfAndOnlyIfPequiv1Mod42006-10-06 13:53:172006-10-07 10:49:15Theoremquadratic residue% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}\begin{thm}
Let $p$ be an odd prime. Then $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1 \mod 4$.
\end{thm}
\begin{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 $p\equiv 3 \mod 4$.
\end{proof}