First Supplement to Law of Quadratic Reciprocity

Theorem

 * $\left({\dfrac {-1} p}\right) = \left({-1}\right)^{\left({p - 1}\right) / 2} = \begin{cases}

+1 & : p \equiv 1 \pmod 4 \\ -1 & : p \equiv 3 \pmod 4 \end{cases}$ where $\left({\dfrac {-1} p}\right)$ is defined as the Legendre symbol.

Proof
From Euler's Criterion, and the definition of the Legendre symbol, we have that:
 * $\left({\dfrac a p}\right) \equiv a^{\left({\frac {p - 1} 2}\right)} \pmod p$

The result follows by putting $a = -1$.

Also see

 * Law of Quadratic Reciprocity
 * Second Supplement to Law of Quadratic Reciprocity