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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.)

Keywords

Chinese Remainder Theorem Congruence Class Legendre Symbol Primitive Root Modulo Jacobi Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

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