Quadratic Reciprocity

In this chapter we describe a procedure for deciding efficiently whether or not a given number is a square modulo m. The main result, known as the law of quadratic reciprocity, was first proved by Gauss (1801) and is a cornerstone of number theory. The last section gives some applications of quadratic reciprocity to primality testing. In the next chapter we give a variety of other applications of the ideas in this chapter.

A curious aspect of the main result of this chapter is that while quadratic reciprocity makes it easy to determine whether or not the congruence x ² ≡ a (mod m) has a solution, the method provides very little help in determining what number is a solution if there is one. (See Section 22C for some implications of this fact.)