Quadratic Reciprocity Theorem

Quadratic reciprocity is a classic result of number theory.<\br> It is one of the most important theorems in the study of quadratic residues.

It states that $\left(\frac{p}{q}\right)= \left(\frac{q}{p}\right)$ for primes $p$ and $q$ greater than $2$ where both are not of the form $4n+3$ for some integer $n$ . If both $p$ and $q$ are of the form $4n+3$ , then $\left(\frac{p}{q}\right)= -\left(\frac{q}{p}\right)$

Another way to state this is:<\br> $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}$

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Quadratic Reciprocity Theorem