The Quadratic Gauss Sum Redux

Let p be an odd prime and ζ be a primitive p-root of unity. For any integer a prime to p, let ( p ) denote the Legendre symbol, which is 1 if a is a square mod p, and is −1 otherwise. Using Euler’s Criterion that a(p−1)/2 = ( p ) mod p, it follows that the Legendre symbol gives a homomorphism from the multiplicative group of nonzero elements Fp of Fp = Z/pZ to {±1}. Gauss’s law of quadratic reciprocity states that for any other odd prime q, ( q p )( p q ) = (−1). A table describing the multitude of proofs of this cherished result over the past two centuries is given in Appendix B of [10], which shows that the starting point of many of the proofs (including one of Gauss’s) is the quadratic Gauss sum,