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[parent] cases when minus one is a quadratic residue (Theorem)
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
$\displaystyle \left(\frac{-1}{p}\right)\equiv (-1)^{(p-1)/2} \mod p.$
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$. $ \qedsymbol$



"cases when minus one is a quadratic residue" is owned by alozano.
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See Also: Euler's criterion, values of the Legendre symbol


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Cross-references: even, integer, Euler's criterion, Legendre symbol, congruent, quadratic residue, prime, odd

This is version 3 of cases when minus one is a quadratic residue, born on 2006-10-06, modified 2006-10-07.
Object id is 8424, canonical name is 1IsQuadraticResidueIfAndOnlyIfPequiv1Mod4.
Accessed 752 times total.

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AMS MSC: 11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity)
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