First Supplement to Law of Quadratic Reciprocity

Theorem

 * $\paren {\dfrac {-1} p} = \paren {-1}^{\paren {p - 1} / 2} = \begin{cases}

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

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

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

Also see

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