The Quadratic Gauss Sum Redux

Abstract

Let p be an odd prime and ζ be a primitive pth-root of unity. For any integer a prime to p, let denote the Legendre symbol, which is 1 if a is a square mod p, and is -1 otherwise. Using Euler's Criterion that 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,