Editing Quadratic reciprocity (section)

===Eisenstein integers===
Consider the following third root of unity:

:<math>\omega = \frac{-1 + \sqrt{-3}}{2}=e^\frac{2\pi \imath}{3}.</math>

The ring of Eisenstein integers is <math>\Z[\omega].</math><ref>See the articles on [[Eisenstein integer]] and [[cubic reciprocity]] for definitions and notations.</ref> For an Eisenstein prime <math>\pi, \mathrm{N} \pi \neq 3,</math> and an Eisenstein integer <math>\alpha</math> with <math>\gcd(\alpha, \pi) = 1,</math> define the quadratic character for <math>\Z[\omega]</math> by the formula

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

Let λ = ''a'' + ''bω'' and μ = ''c'' + ''dω'' be distinct Eisenstein primes where ''a'' and ''c'' are not divisible by 3 and ''b'' and ''d'' are divisible by 3. Eisenstein proved<ref>Lemmermeyer, Thm. 7.10, p. 217</ref>

:<math> \left[\frac{\lambda}{\mu}\right]_2 \left [\frac{\mu}{\lambda}\right ]_2 = (-1)^{\frac{\mathrm{N} \lambda - 1}{2}\frac{\mathrm{N} \mu-1}{2}}, \qquad \left [\frac{1-\omega}{\lambda}\right ]_2 =\left(\frac{a}{3}\right), \qquad \left [\frac{2}{\lambda}\right ]_2 =\left (\frac{2}{\mathrm{N} \lambda }\right).</math>