Quadratic Reciprocity

Part of the Undergraduate Texts in Mathematics book series (UTM)

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 x2a (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.)


Chinese Remainder Theorem Congruence Class Legendre Symbol Primitive Root Modulo Jacobi Symbol 
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© Springer-Verlag Berlin Heidelberg 2009

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