Editing Quadratic reciprocity (section)

== Other rings ==
There are also quadratic reciprocity laws in [[ring (mathematics)|ring]]s other than the integers.

===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>

===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>

===Imaginary quadratic fields===
The above laws are special cases of more general laws that hold for the [[ring of integers]] in any [[Quadratic field|imaginary quadratic number field]]. Let ''k'' be an imaginary quadratic number field with ring of integers <math>\mathcal{O}_k.</math> For a [[Number field#Prime ideals|prime ideal]] <math>\mathfrak{p} \subset \mathcal{O}_k </math> with odd norm <math>\mathrm{N} \mathfrak{p}</math> and <math>\alpha\in \mathcal{O}_k,</math> define the quadratic character for <math>\mathcal{O}_k </math> as

:<math>\left[\frac{\alpha}{\mathfrak{p} }\right]_2 \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} - 1}{2}} \bmod{\mathfrak{p}} = 
\begin{cases}
1 &\alpha\not\in \mathfrak{p} \text{ and } \exists \eta \in \mathcal{O}_k \text{ such that } \alpha - \eta^2 \in \mathfrak{p} \\
-1 & \alpha\not\in \mathfrak{p} \text{ and there is no such } \eta \\
0 & \alpha\in \mathfrak{p}
\end{cases}</math>

for an arbitrary ideal <math>\mathfrak{a} \subset \mathcal{O}_k</math> factored into prime ideals <math>\mathfrak{a} = \mathfrak{p}_1 \cdots \mathfrak{p}_n</math> define

:<math>\left [\frac{\alpha}{\mathfrak{a}}\right ]_2 = \left[\frac{\alpha}{\mathfrak{p}_1 }\right]_2\cdots \left[\frac{\alpha}{\mathfrak{p}_n }\right]_2,</math>

and for <math>\beta \in \mathcal{O}_k</math> define

:<math>\left [\frac{\alpha}{\beta}\right ]_2 = \left [\frac{\alpha}{\beta \mathcal{O}_k}\right ]_2. </math>

Let <math>\mathcal{O}_k = \Z \omega_1\oplus \Z \omega_2,</math> i.e. <math>\left\{\omega_1, \omega_2\right\}</math> is an [[Number field#Integral basis|integral basis]] for <math>\mathcal{O}_k.</math> For <math>\nu \in \mathcal{O}_k</math> with odd norm <math>\mathrm{N}\nu,</math> define (ordinary) integers ''a'', ''b'', ''c'', ''d'' by the equations,

:<math>\begin{align}
\nu\omega_1&=a\omega_1+b\omega_2\\
\nu\omega_2&=c\omega_1+d\omega_2
\end{align}</math>

and a function

:<math>\chi(\nu) := \imath^{(b^2-a+2)c+(a^2-b+2)d+ad}.</math>

If ''m'' = ''Nμ'' and ''n'' = ''Nν'' are both odd, Herglotz proved<ref>Lemmermeyer, Thm 8.15, p.256 ff</ref>

:<math> \left [\frac{\mu}{\nu}\right ]_2 \left[\frac{\nu}{\mu}\right]_2 = (-1)^{\frac{m-1}{2}\frac{n-1}{2}} \chi(\mu)^{m\frac{n-1}{2}} \chi(\nu)^{-n\frac{m-1}{2}}. </math>

Also, if

:<math> \mu \equiv\mu' \bmod{4} \quad \text{and} \quad \nu \equiv\nu' \bmod{4}</math>

Then<ref>Lemmermeyer Thm. 8.18, p. 260</ref>

:<math> \left [\frac{\mu}{\nu}\right ]_2 \left[\frac{\nu}{\mu}\right]_2 = \left [\frac{\mu'}{\nu'}\right ]_2 \left[\frac{\nu'}{\mu'}\right]_2.</math>

===Polynomials over a finite field===
Let ''F'' be a [[finite field]] with ''q'' = ''p<sup>n</sup>'' elements, where ''p'' is an odd prime number and ''n'' is positive, and let ''F''[''x''] be the [[Polynomial ring|ring of polynomials]] in one variable with coefficients in ''F''. If <math>f,g \in F[x]</math> and ''f'' is [[Irreducible polynomial|irreducible]], [[Monic polynomial|monic]], and has positive degree, define the quadratic character for ''F''[''x''] in the usual manner:

:<math>\left(\frac{g}{f}\right) = \begin{cases}
1 & \gcd(f,g)=1 \text{ and } \exists h,k \in F[x] \text{ such that }g-h^2 = kf \\
-1 & \gcd(f,g)=1 \text{ and } g \text{ is not a square}\bmod{f}\\
0 & \gcd(f,g)\ne 1
\end{cases} </math>

If <math>f=f_1 \cdots f_n</math> is a product of monic irreducibles let

:<math>\left(\frac{g}{f}\right) = \left(\frac{g}{f_1}\right) \cdots \left(\frac{g}{f_n}\right). </math>

Dedekind proved that if <math>f,g \in F[x]</math> are monic and have positive degrees,<ref>Bach & Shallit, Thm. 6.7.1</ref>

:<math>\left(\frac{g}{f}\right) \left(\frac{f}{g}\right) = (-1)^{\frac{q-1}{2}(\deg f)(\deg g)}.</math>