Английская Википедия:Eisenstein integer

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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

<math>z = a + b\omega ,</math>

where Шаблон:Math and Шаблон:Math are integers and

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

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

Файл:Eisenstein integer lattice.png

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

<math>z^2 - (2a - b)\;\!z + \left(a^2 - ab + b^2\right)~.</math>

In particular, Шаблон:Math satisfies the equation

<math>\omega^2 + \omega + 1 = 0~.</math>

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

<math>(a + b\;\!\omega) \;\! (c + d\;\!\omega)=(ac - bd) + (bc + ad - bd)\;\!\omega~.</math>

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

<math>{\left|a + b\;\!\omega\right|}^2 \,= \, {(a - \tfrac{1}{2} b)}^2 + \tfrac{3}{4} b^2 \, = \, a^2 - ab + b^2~,</math>

which is clearly a positive ordinary (rational) integer.

Also, the complex conjugate of Шаблон:Math satisfies

<math>\bar\omega = \omega^2~.</math>

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

Euclidean domain

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

<math>N(a+b\,\omega) = a^2 - a b + b^2. </math>

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

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

Here, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон: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 Шаблон:Math:

<math> \frac{\alpha}{\beta}\ =\ \tfrac{1}{\ |\beta|^2}\alpha\overline{\beta} \ =\ a+bi \ =\ a+\tfrac{1}{\sqrt3}b+\tfrac{2}{\sqrt3}b\omega,</math>

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

<math>\kappa = \left\lfloor a+\tfrac{1}{\sqrt3}b\right\rceil + \left\lfloor \tfrac{2}{\sqrt3}b\right\rceil\omega \ \ \text{ and }\ \ \rho = {\alpha} - \kappa\beta.</math>

Here <math>\lfloor x\rceil</math> may denote any of the standard rounding-to-integer functions.

The reason this satisfies Шаблон:Math, while the analogous procedure fails for most other quadratic integer rings, is as follows. A fundamental domain for the ideal Шаблон:Math, acting by translations on the complex plane, is the 60°–120° rhombus with vertices Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math. Any Eisenstein integer Шаблон:Math lies inside one of the translates of this parallelogram, and the quotient Шаблон:Math is one of its vertices. The remainder is the square distance from Шаблон:Math to this vertex, but the maximum possible distance in our algorithm is only <math>\tfrac{\sqrt3}2 |\beta|</math>, so <math>|\rho| \leq \tfrac{\sqrt3}2 |\beta|< |\beta|</math>. (The size of Шаблон:Math could be slightly decreased by taking Шаблон:Math to be the closest corner.)

Eisenstein primes

Файл:EisensteinPrimes-01.svg

Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form Шаблон:Math. All others have an absolute value squared equal to a natural prime.

Файл:Eisenstein primes.svg

Eisenstein primes in a larger range

If Шаблон:Math and Шаблон:Math are Eisenstein integers, we say that Шаблон:Math divides Шаблон:Math if there is some Eisenstein integer Шаблон:Math such that Шаблон:Math. A non-unit Eisenstein integer Шаблон:Math is said to be an Eisenstein prime if its only non-unit divisors are of the form Шаблон:Math, where Шаблон: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. First, an ordinary prime number (or rational prime) which is congruent to Шаблон:Math is also an Eisenstein prime. Second, Шаблон:Math and each rational prime congruent to Шаблон:Math are equal to the norm Шаблон:Math of an Eisentein integer Шаблон:Math. Thus, such a prime may be factored as Шаблон:Math, 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 Шаблон:Math are:

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

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

Шаблон:Math

Шаблон:Math.

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

Шаблон:Math.

Some non-real Eisenstein primes are

Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math.

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

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

Eisenstein series

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

Quotient of Шаблон:Math by the Eisenstein integers

The quotient of the complex plane Шаблон:Math by the lattice containing all Eisenstein integers is a complex torus of real dimension Шаблон:Math. This is one of two tori with maximal symmetry among all such complex tori.Шаблон:Citation needed 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 Шаблон:Math.)

See also

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

External links

• Eisenstein Integer--from MathWorld