Eisenstein integer

In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form 
 
where  and  are integers and
 
is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.

Properties 
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field  – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each  is a root of the monic polynomial
 
In particular,  satisfies the equation
 

The product of two Eisenstein integers  and  is given explicitly by
 

The 2-norm of an Eisenstein integer is just its squared modulus, and is given by 
 
which is clearly a positive ordinary (rational) integer.

Also, the complex conjugate of  satisfies
 

The group of units in this ring is the cyclic group formed by the sixth roots of unity in the complex plane: , the Eisenstein integers of norm .

Euclidean domain 
The ring of Eisenstein integers forms a Euclidean domain whose norm  is given by the square modulus, as above:
 

A division algorithm, applied to any dividend  and divisor , gives a quotient  and a remainder  smaller than the divisor, satisfying:
 

Here, , , ,  are all Eisenstein integers. This algorithm implies the Euclidean algorithm, which proves Euclid's lemma and the unique factorization of Eisenstein integers into Eisenstein primes.

One division algorithm is as follows.  First perform the division in the field of complex numbers, and write the quotient in terms of :
 
for rational . Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:
 
Here  may denote any of the standard rounding-to-integer functions.

The reason this satisfies , while the analogous procedure fails for most other quadratic integer rings, is as follows. A fundamental domain for the ideal , acting by translations on the complex plane, is the 60°–120° rhombus with vertices , , , . Any Eisenstein integer  lies inside one of the translates of this parallelogram, and the quotient  is one of its vertices. The remainder is the square distance from  to this vertex, but the maximum possible distance in our algorithm is only , so . (The size of  could be slightly decreased by taking  to be the closest corner.)

Eisenstein primes 

If  and  are Eisenstein integers, we say that  divides  if there is some Eisenstein integer  such that . A non-unit Eisenstein integer  is said to be an Eisenstein prime if its only non-unit divisors are of the form , where  is any of the six units. They are the corresponding concept to the Gaussian primes in the Gaussian integers.

There are two types of Eisenstein prime. First, an ordinary prime number (or rational prime) which is congruent to  is also an Eisenstein prime. Second,  and each rational prime congruent to  are equal to the norm  of an Eisentein integer . Thus, such a prime may be factored as , and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.

The first few Eisenstein primes of the form  are:
 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... .

Natural primes that are congruent to  or  modulo  are not Eisenstein primes: they admit nontrivial factorizations in . For example:
 
 .

In general, if a natural prime  is  modulo  and can therefore be written as , then it factorizes over  as
 .

Some non-real Eisenstein primes are
 , , , , , , .

Up to conjugacy and unit multiples, the primes listed above, together with  and , are all the Eisenstein primes of absolute value not exceeding .

, the largest known real Eisenstein prime is the tenth-largest known prime , discovered by Péter Szabolcs and PrimeGrid. With one exception, all larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to , and all Mersenne primes greater than  are congruent to ; thus no Mersenne prime is an Eisenstein prime.

Eisenstein series 
The sum of the reciprocals of all Eisenstein integers excluding  raised to the sixth power can be expressed in terms of the gamma function:

where  are the Eisenstein integers and  is the Eisenstein series of weight 6.

Quotient of  by the Eisenstein integers 
The quotient of the complex plane  by the lattice containing all Eisenstein integers is a complex torus of real dimension . This is one of two tori with maximal symmetry among all such complex tori.  This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. (The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of Gaussian integers, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as .)

See also 
 Gaussian integer
 Cyclotomic field
 Systolic geometry
 Hermite constant
 Cubic reciprocity
 Loewner's torus inequality
 Hurwitz quaternion
 Quadratic integer
 Dixon elliptic functions

Notes

External links 
 Eisenstein Integer--from MathWorld

Algebraic numbers
Quadratic irrational numbers
Cyclotomic fields
Lattice points
Systolic geometry
Integers