Number Theory and its Applications

9 Theory of Quadratic Residues

“Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.”

– Johann Wolfgang von Goethe

9.1 Introduction

In the chapter Fermat’s little theorem, we defined quadratic congruence equation of the form ax² + bx + c ≡ 0(mod p), p being an odd prime and a ≢ 0( mod p). There we have discoursed the solution of the quadratic congruence of the type x² + 1 ≡ (mod p). Also, how to solve a quadratic congruence of that form, was being treated there. Following the path of solving a quadratic congruence, we need an important fact of modern number theory, known as quadratic reciprocity law, a major contribution of Gauss in 1796. After ...