A Proof of the Law of Reciprocity for Jacobi Symbols

The congruence x 2 ≡ a (mod p ), where p is an odd prime and a is any number not divisible by p , sometimes has a solution, but sometimes it has not. Gauss’s symbol ( a/p ), also known as Legendre’s symbol, is defined as ( a/p ) = l if there exists at least one x as a solution to this congruence and ( a/p ) = − l if no such solution exists. The “Law of Reciprocity”, the famous theorem by Gauss, then states that, if p and q are odd primes, where