Quadratic Reciprocity in Number Theory and Group Theory

Abstract

The purpose of the project is to present several proofs and applications of quadratic reciprocity for integers.We present four different proofs of quadratic reci- procity. We start by discussing Gauss lemma and Euler’s critertion in Number theory. Then we reformulate Gauss lemma and Euler’s critertion in terms of Group Theory, which leads to a group theory proof of quadratic reciprocity. Also we intro- duce Gauss sums and apply them for another proof of quadratic reciprocity. Finally we use discrete Fourier analysis on cyclic groups to give the fourth proof of quadratic reciprocity over Z .