Quadratic Reciprocity in Number Theory and Group Theory

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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 .
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