Quadratic Residues and the Quadratic Reciprocity Law

Abstract

As shown in Chapter 5, the problem of solving a polynomial congruence

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

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

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

((1))

where 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.