Law of quadratic reciprocity states as follows:
Law of quadratic reciprocity — Let $p$ and $q$ be distinct odd prime numbers, and define the Legendre symbol as:
$$ \left(\frac {q}{p}\right)=\left\{\begin{array}{rl} 1 & \text{if } n^2\equiv q \pmod p \text{ for some integer } n, \\ -1 & \text{otherwise.} \end{array} \right.$$
Then:
$${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}$$
I have heard from class that there are hundreds of proof of this theorem. But the proof that I have learned in class is a very elementary one. As we know that many theorems in number theory have some very nice explanations using abstract algebra. Is there a proof of this theorem from the perspective of abstract algebra? And what is the intuition behind it? Thank you!