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

$z=a+b\omega ,$

where Template:Math and Template:Math are integers and

$\omega =\frac{-1+i\sqrt{3}}{2}=e^{i2\pi /3}$

is a primitive (hence non-real) cube root of unity.

Eisenstein integers as the points of a certain triangular lattice in the complex plane

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 Template:Math – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each Template:Math is a root of the monic polynomial

$z^2-(2a-b)z+(a^2-ab+b^2).$

In particular, Template:Math satisfies the equation

${\omega }^2+\omega +1=0.$

The product of two Eisenstein integers Template:Math and Template:Mvar is given explicitly by

$(a+b\omega )(c+d\omega )=(ac-bd)+(bc+ad-bd)\omega .$

The 2-norm of an Eisenstein integer is just its squared modulus, and is given by

${|a+b\omega |}^2={(a-\frac{1}{2}b)}^2+\frac{3}{4}{b}^2=a^2-ab+b^2,$

which is clearly a positive ordinary (rational) integer.

Also, the complex conjugate of Template:Math satisfies

$\overline{\omega }={\omega }^2.$

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

Euclidean domain

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

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

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

$\alpha =\kappa \beta +\rho \text{ with }N(\rho )<N(\beta ).$

Here, Template:Math, Template:Math, Template:Math, Template:Math 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 Template:Math:

$\frac{\alpha }{\beta }=\frac{1}{|\beta {|}^2}\alpha \bar{\beta }=a+bi=a+\frac{1}{\sqrt{3}}b+\frac{2}{\sqrt{3}}b\omega ,$

for rational Template:Math. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:

$\kappa =\lfloor a+\frac{1}{\sqrt{3}}b\rceil +\lfloor \frac{2}{\sqrt{3}}b\rceil \omega \text{ and }\rho =\alpha -\kappa \beta .$

Here $\lfloor x\rceil$ may denote any of the standard rounding-to-integer functions.

The reason this satisfies Template:Math, while the analogous procedure fails for most other quadratic integer rings, is as follows. A fundamental domain for the ideal Template:Math, acting by translations on the complex plane, is the 60°–120° rhombus with vertices Template:Math, Template:Math, Template:Math, Template:Math. Any Eisenstein integer Template:Math lies inside one of the translates of this parallelogram, and the quotient Template:Math is one of its vertices. The remainder is the square distance from Template:Math to this vertex, but the maximum possible distance in our algorithm is only $\frac{\sqrt{3}}{2}|\beta |$, so $|\rho |\le \frac{\sqrt{3}}{2}|\beta |<|\beta |$. (The size of Template:Math could be slightly decreased by taking Template:Math to be the closest corner.)

Eisenstein primes

Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form Template:Math. All others have an absolute value equal to 3 or square root of a natural prime of the form Template:Math.

Eisenstein primes in a larger range

If Template:Math and Template:Math are Eisenstein integers, we say that Template:Math divides Template:Math if there is some Eisenstein integer Template:Math such that Template:Math. A non-unit Eisenstein integer Template:Math is said to be an Eisenstein prime if its only non-unit divisors are of the form Template:Math, where Template:Math 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.

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

In the second type, factors of Template:Math, $1-\omega$ and $1-{\omega }^2$ are associates: $1-\omega =(-\omega )(1-{\omega }^2)$, so it is regarded as a special type in some books.

The first few Eisenstein primes of the form Template:Math are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... Template:OEIS.

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

Template:Math

Template:Math.

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

Template:Math.

Some non-real Eisenstein primes are

Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math.

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

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

Eisenstein series

The sum of the reciprocals of all Eisenstein integers excluding Template:Math raised to the fourth power is Template:Math: $$\sum _{z\in 𝐄\setminus \{0\}}\frac{1}{z^4}=G_4\left(e^{\frac{2\pi i}{3}}\right)=0$$ so $e^{2\pi i/3}$ is a root of j-invariant. In general $G_k\left(e^{\frac{2\pi i}{3}}\right)=0$ if and only if $k\equiv ̸0(\mathrm{mod}6)$.

The sum of the reciprocals of all Eisenstein integers excluding Template:Math raised to the sixth power can be expressed in terms of the gamma function: $$\sum _{z\in 𝐄\setminus \{0\}}\frac{1}{z^6}=G_6\left(e^{\frac{2\pi i}{3}}\right)=\frac{\mathrm{\Gamma }(1/3{)}^{18}}{8960{\pi }^6}$$ where Template:Math are the Eisenstein integers and Template:Math is the Eisenstein series of weight 6.

Quotient of Template:Math by the Eisenstein integers

The quotient of the complex plane Template:Math by the lattice containing all Eisenstein integers is a complex torus of real dimension Template:Math. 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.

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 Template:Math.

See also

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

External links

• Eisenstein Integer--from MathWorld