Robert Hines October 16, 2015
Contents
1 Proofs Using the Quadratic Gauss Sum 1
2 Some Related Lemmata and a Few More Proofs 3
3 A Proof Using Jacobi Sums 6
4 The Quadratic Character of 2 and −1 7
5 Appendix: Sign of the Quadratic Gauss Sum 8
1 Proofs Using the Quadratic Gauss Sum
Definition. A Gauss sum g(a, χ) associated to a character χ of modulus n (a homomor- phism χ :(Z/nZ)× → C extended to Z, χ(k) = 0 if (k, n) > 1) is
n−1 X g(a, χ) = χ(k)e2πiak/n k=1 Observe that for (a, n) = 1
n−1 n−1 X X g(a, χ) = χ(k)e2πiak/n = χ(l)¯χ(a)e2πil/n =χ ¯(a)g(1, χ) k=1 l=1 with l = ka. Also observe that X g(a, χ) = χ¯(k)e−2πiak/n =χ ¯(−1)g(a, χ¯). k Proposition. For a non-principal character χ of prime modulus p and (a, p) = 1 we have |g(a, χ)|2 = p.
1 P Proof. We evaluate a g(a, χ)g(a, χ) two different ways: X X X X g(a, χ)g(a, χ) = χ(kl−1)e2πia(k−l)/p = χ(kl−1) e2πia(k−l)/p a a,k,l k,l a X −1 = χ(kl )pδkl = p(p − 1), k,l
X X g(a, χ)g(a, χ) = χ¯(a)g(1, χ)χ(a)g(1, χ) a a = (p − 1)|g(1, χ)|2.
For χ real, the above says g(1, χ)2 = χ(−1)p, which we will use below. Theorem (Quadratic Reciprocity). For odd primes p, q we have p q p−1 q−1 = (−1) 2 (−1) 2 q p