Gauss's lemma (number theory)

Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.

It made its first appearance in Carl Friedrich Gauss's third proof (1808) of quadratic reciprocity and he proved it again in his fifth proof (1818).

In the original statement, <math>I=\{1,2,\dots,\frac{p-1}2\}</math>.

The proof is almost the same.

Applications

Gauss's lemma is used in many, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of in , the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.