# 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:

$$\pm 1\mathit{\hspace{1em}\hspace{1em}}\pm \omega \mathit{\hspace{1em}\hspace{1em}}\pm {\omega}^{2}$$ |

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 $q\equiv 2\phantom{\rule{veryverythickmathspace}{0ex}}(mod3)$ 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 $N(q)\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod3)$.

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 $N(q)\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod3)$ is prime in $\mathbb{Z}$ and

$a$ | $\equiv $ | $2\phantom{\rule{veryverythickmathspace}{0ex}}(mod3)$ | ||

$b$ | $\equiv $ | $0\phantom{\rule{veryverythickmathspace}{0ex}}(mod3).$ |

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 \ne \pi $. If $\alpha $ is any
element of $K$ such that $\rho \nmid \alpha $, then

$${\alpha}^{N(\rho )-1}\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod\rho ).$$ |

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

$${\chi}_{\rho}:K\to \{1,\omega ,{\omega}^{2}\}$$ |

by

${\chi}_{\rho}(\alpha )$ | $\equiv $ | ${\alpha}^{(N(\rho )-1)/3}\text{if}\rho \nmid \alpha $ | ||

${\chi}_{\rho}(\alpha )$ | $=$ | $0\text{if}\rho \mid \alpha .$ |

${\chi}_{\rho}$ 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

$m$ | $=$ | $(\rho +1)/3\mathit{\hspace{1em}\hspace{1em}}\text{if}\rho \text{is a rational prime}$ | ||

$m$ | $=$ | $(a+1)/3\mathit{\hspace{1em}\hspace{1em}}\text{if}\rho =a+b\omega \text{is a complex prime.}$ |

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 $x\equiv 1\phantom{\rule{veryverythickmathspace}{0ex}}(mod4)$. 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 |
---|---|

Canonical name | CubicReciprocityLaw |

Date of creation | 2013-03-22 13:41:26 |

Last modified on | 2013-03-22 13:41:26 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Topic |

Classification | msc 11A15 |

Related topic | QuadraticReciprocityRule |

Defines | cubic residue |

Defines | cubic residue character |