Quadratic reciprocity via discrete Fourier analysis

In this post, we give a proof of the law of quadratic reciprocity based on discrete Fourier analysis and more precisely Gauss sums. This celebrated theorem has numerous proofs, but the most elementary ones usually boil down to some tedious combinatorial computations. In particular, after one has derived Gauss' lemma or some similar statement that is usually used in the proof, the big picture may become unclear, and one is left with some combinatorial counting problem. On the other hand, if one uses discrete Fourier analysis to prove the reciprocity law, the structure of the proof is simple, and one works in the context of trigonometric sums instead of solving an isolated combinatorial problem. Moreover, this method generalizes to higher reciprocity laws as well, and similar ideas as in what follows can be applied for example in proving the functional equation of the $L$-functions, the class number formula or the Pólya-Vinogradov inequality. Some of these results will be discussed in later posts.