A Concrete Introduction to Higher Algebra pp 433-457 | Cite as

# 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.)

## Keywords

Chinese Remainder Theorem Congruence Class Legendre Symbol Primitive Root Modulo Jacobi Symbol## Preview

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