Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, … See more

The supplements provide solutions to specific cases of quadratic reciprocity. They are often quoted as partial results, without having to resort to the complete theorem.
Trivially 1 is a quadratic residue for all primes. Th… See more

Apparently, the shortest known proof yet was published by B. Veklych in the American Mathematical Monthly.
The value of the Legendre symbol of (used in the proof above) follows directly from Euler's crite… See more

The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol.
In this article p and q always refer to distinct positive od… See more

The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic r… See more

There are also quadratic reciprocity laws in rings other than the integers.
In his second monograph on quartic reciprocity Gauss stated quadratic reciprocity for the ring of Gaussian integers, saying that … See more