Calculation on Quadratic Reciprocity Law and the Quadratic Gauss Sums

Quadratic reciprocity law is most famous theorem in elementary number theory, found by famous French Mathematician Legendre. The famous German Mathematician Gauss called it the yeast of number and gave first complete proof for it, which have been developed afterwards by many mathematicians all over world. In process of development and proof of quadratic reciprocity law, quadratic Gauss sums had played very important role. On other hand, quadratic Gauss sums are a special kind of character sums which are very important tools in study of number theory with many applications in a series of important problems. In this paper, authors have given a new method of calculating quadratic Gauss sums by using method from linear algebra. The main idea is to construct a matrix of which trace is just quadratic Gauss sums thus that could be calculated if corresponding characteristic values of matrix could be given by linear algebra method.