# cubic reciprocity law

In a ring $\mathbb{Z}/n\mathbb{Z}$, a cubic residue is just a value of the function $x^{3}$ for some invertible element $x$ of the ring. Cubic residues display a reciprocity phenomenon similar to that seen with quadratic residues. But we need some preparation in order to state the cubic reciprocity law.

$\omega$ will denote $\frac{-1+i\sqrt{3}}{2}$, which is one of the complex cube roots of $1$. $K$ will denote the ring $K=\mathbb{Z}[\omega]$. The elements of $K$ are the complex numbers $a+b\omega$ where $a$ and $b$ are integers. We define the norm $N:K\to\mathbb{Z}$ by

 $N(a+b\omega)=a^{2}-ab+b^{2}$

or equivalently

 $N(z)=z\overline{z}\;.$

Whereas $\mathbb{Z}$ has only two units (meaning invertible elements), namely $\pm 1$, $K$ has six, namely all the sixth roots of 1:

 $\pm 1\qquad\pm\omega\qquad\pm\omega^{2}$

and we know $\omega^{2}=-1-\omega$. Two nonzero elements $\alpha$ and $\beta$ of $K$ are said to be associates if $\alpha=\beta\mu$ for some unit $\mu$. This is an equivalence relation, and any nonzero element has six associates.

$K$ is a principal ring, hence has unique factorization. Let us call $\rho\in K$irreducible” if the condition $\rho=\alpha\beta$ implies that $\alpha$ or $\beta$, but not both, is a unit. It turns out that the irreducible elements of $K$ are (up to multiplication by units):

– the number $1-\omega$, which has norm 3. We will denote it by $\pi$.

positive real integers $q\equiv 2\pmod{3}$ which are prime in $\mathbb{Z}$. Such integers are called rational primes in $K$.

– complex numbers $q=a+b\omega$ where $N(q)$ is a prime in $Z$ and $N(q)\equiv 1\pmod{3}$.

For example, $3+2\omega$ is a prime in $K$ because its norm, 7, is prime in $\mathbb{Z}$ and is 1 mod 3; but 7 is not a prime in $K$.

Now we need some convention whereby at most one of any six associates is called a prime. By convention, the following numbers are nominated:

– the number $\pi$.

– rational primes (rather than their negative or complex associates).

– complex numbers $q=a+b\omega$ where $N(q)\equiv 1\pmod{3}$ is prime in $\mathbb{Z}$ and

 $\displaystyle a$ $\displaystyle\equiv$ $\displaystyle 2\pmod{3}$ $\displaystyle b$ $\displaystyle\equiv$ $\displaystyle 0\pmod{3}\;.$

One can verify that this selection exists and is unambigous.

Next, we seek a three-valued function analogous to the two-valued quadratic residue character $x\mapsto\left(\frac{x}{p}\right)$. Let $\rho$ be a prime in $K$, with $\rho\neq\pi$. If $\alpha$ is any element of $K$ such that $\rho\nmid\alpha$, then

 $\alpha^{N(\rho)-1}\equiv 1\pmod{\rho}\;.$

Since $N(\rho)-1$ is a multiple of 3, we can define a function

 $\chi_{\rho}:K\to\{1,\omega,\omega^{2}\}$

by

 $\displaystyle\chi_{\rho}(\alpha)$ $\displaystyle\equiv$ $\displaystyle\alpha^{(N(\rho)-1)/3}\text{ if }\rho\nmid\alpha$ $\displaystyle\chi_{\rho}(\alpha)$ $\displaystyle=$ $\displaystyle 0\text{ if }\rho\mid\alpha\;.$

$\chi_{\rho}$ is a character, called the cubic residue character mod $\rho$. We have $\chi_{\rho}(\alpha)=1$ if and only if $\alpha$ is a nonzero cube mod $\rho$. (Compare Euler’s criterion.)

At last we can state this famous result of Eisenstein and Jacobi:

Theorem (Cubic Reciprocity Law): If $\rho$ and $\sigma$ are any two distinct primes in $K$, neither of them $\pi$, then

 $\chi_{\rho}(\sigma)=\chi_{\sigma}(\rho)\;.$

The quadratic reciprocity law has two “supplements” which describe $\left(\frac{-1}{p}\right)$ and $\left(\frac{2}{p}\right)$. Likewise the cubic law has this supplement, due to Eisenstein:

Theorem: For any prime $\rho$ in $K$, other than $\pi$,

 $\chi_{\rho}(\pi)=\omega^{2m}$

where

 $\displaystyle m$ $\displaystyle=$ $\displaystyle(\rho+1)/3\qquad\text{ if \rho is a rational prime}$ $\displaystyle m$ $\displaystyle=$ $\displaystyle(a+1)/3\qquad\text{ if \rho=a+b\omega is a complex prime.}$

Remarks: Some writers refer to our “irreducible” elements as “primes” in $K$; what we have called primes, they call “primary primes”.

The quadratic reciprocity law would take a simpler form if we were to make a different convention on what is a prime in $\mathbb{Z}$, a convention similar to the one in $K$: a prime in $\mathbb{Z}$ is either 2 or an irreducible element $x$ of $\mathbb{Z}$ such that $x\equiv 1\pmod{4}$. The primes would then be 2, -3, 5, -7, -11, 13, …and the QRL would say simply

 $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=1$

for any two distinct odd primes $p$ and $q$.

Title cubic reciprocity law CubicReciprocityLaw 2013-03-22 13:41:26 2013-03-22 13:41:26 mathcam (2727) mathcam (2727) 7 mathcam (2727) Topic msc 11A15 QuadraticReciprocityRule cubic residue cubic residue character