Dr. Sergey OblezinMALAK SALEH DUGHAILEB ALOTAIBI2022-05-282022-05-28https://drepo.sdl.edu.sa/handle/20.500.14154/38364The 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 .en