The law of quadratic reciprocity

Abstract

Let p and q be two distinct odd primes. Then the Legendre symbols \(\left( {\frac{p}{q}} \right)\) and \(\left( {\frac{q}{p}} \right)\) are defined. Can \(\left( {\frac{q}{p}} \right)\) be determined if \(\left( {\frac{p}{q}} \right)\) is known? Gauss’s Law of quadratic reciprocity shows that that is indeed possible.