cubic reciprocity law

In a ring /n, a cubic residueMathworldPlanetmath is just a value of the function x3 for some invertible element x of the ring. Cubic residues display a reciprocity phenomenon similar to that seen with quadratic residuesMathworldPlanetmath. But we need some preparation in order to state the cubic reciprocity law.

ω will denote -1+i32, which is one of the complex cube roots of 1. K will denote the ring K=[ω]. The elements of K are the complex numbersMathworldPlanetmathPlanetmath a+bω where a and b are integers. We define the norm N:K by


or equivalently


Whereas has only two units (meaning invertible elements), namely ±1, K has six, namely all the sixth roots of 1:

±1  ±ω  ±ω2

and we know ω2=-1-ω. Two nonzero elements α and β of K are said to be associatesMathworldPlanetmath if α=βμ for some unit μ. This is an equivalence relationMathworldPlanetmath, and any nonzero element has six associates.

K is a principal ringMathworldPlanetmath, hence has unique factorizationMathworldPlanetmath. Let us call ρKirreduciblePlanetmathPlanetmath” if the condition ρ=αβ implies that α or β, but not both, is a unit. It turns out that the irreducible elements of K are (up to multiplicationPlanetmathPlanetmath by units):

– the number 1-ω, which has norm 3. We will denote it by π.

positive real integers q2(mod3) which are prime in . Such integers are called rational primes in K.

– complex numbers q=a+bω where N(q) is a prime in Z and N(q)1(mod3).

For example, 3+2ω is a prime in K because its norm, 7, is prime in 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 π.

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

– complex numbers q=a+bω where N(q)1(mod3) is prime in and

a 2(mod3)
b 0(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 characterPlanetmathPlanetmath x(xp). Let ρ be a prime in K, with ρπ. If α is any element of K such that ρα, then


Since N(ρ)-1 is a multipleMathworldPlanetmathPlanetmath of 3, we can define a function



χρ(α) α(N(ρ)-1)/3 if ρα
χρ(α) = 0 if ρα.

χρ is a character, called the cubic residue character mod ρ. We have χρ(α)=1 if and only if α is a nonzero cube mod ρ. (Compare Euler’s criterion.)

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

Theorem (Cubic Reciprocity Law): If ρ and σ are any two distinct primes in K, neither of them π, then


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

Theorem: For any prime ρ in K, other than π,



m = (ρ+1)/3   if ρ is a rational prime
m = (a+1)/3   if ρ=a+bω is a complex prime.

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

The quadratic reciprocity law would take a simpler form if we were to make a different convention on what is a prime in , a convention similar to the one in K: a prime in is either 2 or an irreducible element x of such that x1(mod4). The primes would then be 2, -3, 5, -7, -11, 13, …and the QRL would say simply


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