Our first goal in this chapter is to present Gauss’s sixth proof of his Law of Quadratic Reciprocity. The presentation here follows [32, §3.3] fairly closely, except that our Gauss sums are over the complex numbers, as opposed to ibid. where Gauss sums are considered over a finite field. Later in the chapter we introduce the Jacobi symbol and study its basic properties. The Jacobi symbol will make an appearance in Chapter when we give a proof of the Three Squares Theorem.