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 $p\equiv 3mod4$. ∎

Title	cases when minus one is a quadratic residue
Canonical name	CasesWhenMinusOneIsAQuadraticResidue
Date of creation	2013-03-22 16:18:10
Last modified on	2013-03-22 16:18:10
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11A15
Related topic	EulersCriterion
Related topic	ValuesOfTheLegendreSymbol