Quadratic and cubic reciprocity - what makes it tick?

Quadratic reciprocity was reputedly Gauss's favorite theorem. The closer you look at it, the more mysterious it becomes. The problem is motivated by a simple question: if I fix one prime q, then I want to know for which primes p I can solve the congruence q ≡ x² (mod p)? For example, how can I characterize the set of primes p for which I can solve 7 ≡ x² (mod p)? It turns out that the answer only depends on p mod 7! I'd like to walk through the proofs that appear in Ireland and Rosen: "A Classical Introduction to Modern Number Theory". They use Gauss sums. This approach generalizes well enough to work for cubic and bi-quadratic reciprocity. Gauss sums are also fairly mysterious. They have everything to do with finite Fourier analysis and the theory of group characters. The proofs also entail a fair amount of basic number theory: Lagrange's Theorem, Fermat's Little Theorem, the Chinese Remainder Theorem, linear congruences, primitive roots, and cyclotomic fields. Perhaps this meetup should be considered like an anatomy lesson: we're going to pull apart the first 200 pages of Ireland and Rosen to find out what makes the clock of arithmetic reciprocity tick.