Editing Quadratic reciprocity (section)

===Gaussian integers===
In his second monograph on [[quartic reciprocity]]<ref>Gauss, BQ § 60</ref> Gauss stated quadratic reciprocity for the ring <math>\Z[i]</math> of [[Gaussian integer]]s, saying that it is a corollary of the [[quartic reciprocity|biquadratic law]] in <math>\Z[i],</math> but did not provide a proof of either theorem. [[Peter Gustav Lejeune Dirichlet|Dirichlet]]<ref>Dirichlet's proof is in Lemmermeyer, Prop. 5.1 p.154, and Ireland & Rosen, ex. 26 p. 64</ref> showed that the law in <math>\Z[i]</math> can be deduced from the law for <math>\Z</math> without using quartic reciprocity.

For an odd Gaussian prime <math>\pi</math> and a Gaussian integer <math>\alpha</math> relatively prime to <math>\pi,</math> define the quadratic character for <math>\Z[i]</math> by:

:<math>\left[\frac{\alpha}{\pi}\right]_2 \equiv \alpha^\frac{\mathrm{N} \pi - 1}{2}\bmod{\pi} = 
\begin{cases}
1 & \exists \eta \in \Z[i]: \alpha \equiv \eta^2 \bmod{\pi} \\
-1 & \text{otherwise}
\end{cases} </math>

Let <math>\lambda = a + b i, \mu = c + d i</math> be distinct Gaussian primes where ''a'' and ''c'' are odd and ''b'' and ''d'' are even. Then<ref>Lemmermeyer, Prop. 5.1, p. 154</ref>

:<math> \left [\frac{\lambda}{\mu}\right ]_2 = \left [\frac{\mu}{\lambda}\right ]_2, \qquad \left [\frac{i}{\lambda}\right ]_2 =(-1)^\frac{b}{2}, \qquad \left [\frac{1+i}{\lambda}\right ]_2 =\left(\frac{2}{a+b}\right).</math>