Exercise Set 6:

Quadratic Reciprocity

Math 414, Winter 2010, University of Washington

Due Wednesday, February 17, 2010

This homework assignment is purposely short because you also will have a take-home midterm this coming weekend.

1. Let p ≠ 7 be an odd prime. Use the quadratic reciprocity law to prove that −7 is a perfect square modulo p if and only if p ≡ 1, 2, 4 (mod 7).

2. Let ζ_{p} be a primitive pth root of unity. Are there infinitely many primes p such that the Gauss sum ∑_{n=0}^{p−1} (\frac{n}{p}) ζ_{p}^{n} is a real number?