Introduction to Analytic Number Theory pp 178-203 | Cite as

# Quadratic Residues and the Quadratic Reciprocity Law

Chapter

## Abstract

As shown in Chapter 5, the problem of solving a polynomial congruence can be reduced to polynomial congruences with prime moduli plus a set of linear congruences. This chapter is concerned with quadratic congruences of the form where

$$
f\left( x \right) \equiv 0\left( {\bmod m} \right)
$$

$$
{x^2} \equiv n\;(\,\bmod \,\;p)
$$

(1)

*p*is an odd prime and*n*≢ 0 (mod*p*). Since the modulus is prime we know that (1) has at most two solutions. Moreover, if*x*is a solution so is -*x*, hence the number of solutions is either 0 or 2.## Keywords

Diophantine Equation Quadratic Residue Legendre Symbol Linear Congruence 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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© Springer Science+Business Media New York 1976