Quadratic Reciprocity

Quadratic Reciprocity is arguably the most important theorem taught in an elementary number theory course. Since Gauss’ original 1796 proof (by induction!) appeared, more than 100 different proofs have been discovered. Here I present one proof which is not particularly well-known, due to George Rousseau [2]. (The proof was rediscovered more recently by (then) high-schooler Tim Kunisky [1].) Although not the shortest proof, it is the easiest to remember of all the elementary proofs I have encountered. In particular, it does not rely on Gauss’ Lemma, or lattice counting, or Gauss sums; the only ingredients used in the proof are the Chinese Remainder Theorem, Wilson’s Theorem, and Euler’s Criterion. After proving Quadratic Reciprocity for the case of two odd primes, I’ll show how to derive the ‘supplementary’ laws directly from the classical case.