Applications
Gauss's lemma is used in many, but by no means all, of the known proofs of quadratic reciprocity.
For example, Eisenstein used Gauss's lemma to prove that if p is an odd prime then
and used this formula to prove quadratic reciprocity, (and, by using elliptic rather than circular functions, to prove the cubic and quartic reciprocity laws.)
Kronecker used the lemma to show that
Switching p and q immediately gives quadratic reciprocity.
It is also used in what are probably the simplest proofs of the "second supplementary law"
Read more about this topic: Gauss's Lemma (number Theory)