# cases when minus one is a quadratic residue

###### Theorem.

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 $p\equiv 3\mod 4$. ∎

Title cases when minus one is a quadratic residue CasesWhenMinusOneIsAQuadraticResidue 2013-03-22 16:18:10 2013-03-22 16:18:10 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 11A15 EulersCriterion ValuesOfTheLegendreSymbol