Reciprocity laws

Abstract: 

Given an integer a, we wish to know for which prime integers p the congruence x^2 = a mod p is solvable. An important tool for analysing this is the law of quadratic reciprocity. Gauss was not only the first to give a proof, but produced more than five others over the years in an attempt to generalise the law to higher powers. We will, however, give a particularly elegant proof given by Eisenstein that is based on Gauss' work. We'll then see how to extend the result to the cubic case, and then to higher odd prime powers.