Quadratic reciprocity

Let $p$ and $q$ be distinct odd primes. For an integer $a$ and an odd prime $p$, the Legendre symbol $\left(\frac{a}{p}\right)$ is $1$ if $a$ is a non-zero square modulo $p$, $-1$ if it is not a square, and $0$ if $p$ divides $a$.

Show that
$$\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1{)}^{\frac{p-1}{2}\cdot \frac{q-1}{2}}.$$

Gauss called it the golden theorem and gave several proofs. A great many have followed, and they come from counting, from geometry, from algebra and from analysis.