quality:corruption
Gaussian integers

In his second monograph on quartic reciprocity Gauss stated quadratic reciprocity for the ring Z {\displaystyle \mathbb {Z} } of Gaussian integers, saying that it is a corollary of the biquadratic law in Z, {\displaystyle \mathbb {Z},} but did not provide a proof of either theorem. Dirichlet showed that the law in Z {\displaystyle \mathbb {Z} } can be deduced from the law for Z {\displaystyle \mathbb {Z} } without using quartic reciprocity.

For an odd Gaussian prime π {\displaystyle \pi } and a Gaussian integer α {\displaystyle \alpha } relatively prime to π, {\displaystyle \pi,} define the quadratic character for Z {\displaystyle \mathbb {Z} } by:

Let λ = a + b i, μ = c + d i {\displaystyle \lambda =a+bi,\mu =c+di} be distinct Gaussian primes where a and c are odd and b and d are even. Then