First Supplement to Law of Quadratic Reciprocity

Theorem
$$\left({\frac{-1}{p}}\right) = \left({-1}\right)^{\left({p-1}\right)/2} = \begin{cases} +1 & : p \equiv 1 \pmod {4} \\ -1 & : p \equiv 3 \pmod {4} \end{cases}$$

where $$\left({\frac{-1}{p}}\right)$$ is defined as the Legendre symbol.

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

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