Eisenstein integers

In mathematics, Eisenstein integers (named after Gotthold Eisenstein), also known[1] as Eulerian integers (after Leonhard Euler), are complex numbers of the form

\( z = a + b\omega \,\! \)

where a and b are integers and

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

is a primitive (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.

Properties

The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω). To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial

\( z^2 - (2a - b)z + (a^2 - ab + b^2). \,\! \)

In particular, ω satisfies the equation

\( \omega^2 + \omega + 1 = 0. \,\! \)

The norm of a Eisenstein integer is just the square of its absolute value and is given by

\( |a+b\omega|^2 = a^2 - ab + b^2. \,\! \)

Thus the norm of an Eisenstein integer is always an ordinary (rational) integer. Since

\( 4a^2-4ab+4b^2=(2a-b)^2+3b^2, \,\! \)

the norm of a nonzero Eisenstein integer is positive.

The group of units in the ring of Eisenstein integers is the cyclic group formed by the sixth roots of unity in the complex plane. Specifically, they are

{±1, ±ω, ±ω²}

These are just the Eisenstein integers of norm one.
Eisenstein primes
Main article: Eisenstein prime

If x and y are Eisenstein integers, we say that x divides y if there is some Eisenstein integer z such that y = z x.

This extends the notion of divisibility for ordinary integers. Therefore we may also extend the notion of primality; a non-unit Eisenstein integer x is said to be an Eisenstein prime if its only divisors are of the form ux where u is any of the six units.

It may be shown that an ordinary prime number (or rational prime) which is 3 or congruent to 1 mod 3 is of the form x2 − xy + y2 for some integers x, y and may therefore be factored into (x + ωy)(x + ω²y) and because of that it is not prime in the Eisenstein integers. Ordinary primes congruent to 2 mod 3 cannot be factored in this way and they are primes in the Eisenstein integers as well.

Every Eisenstein integer a + bω whose norm a² − ab + b² is a rational prime is an Eisenstein prime. In fact, every Eisenstein prime is of this form, or is a product of a unit and a rational prime congruent to 2 mod 3.
Euclidean domain

The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by

\( N(a + b\,\omega) = a^2 - a b + b^2. \,\! \)

This can be derived as follows:

\( \begin{align}N(a+b\,\omega) &=|a+b\,\omega|^2\\ &=(a+b\,\omega)(a+b\,\bar\omega)\\ &=a^2 + ab(\omega+\bar\omega) + b^2\\ &=a^2 - ab + b^2\end{align} \)

Quotient of C by the Eisenstein integers

The quotient of the complex plane C by the lattice containing all Eisenstein integers is a complex torus of real dimension 2, with the highest degree of symmetry among all such complex tori.
See also

Gaussian integer
Kummer ring
Systolic geometry
Hermite constant
Cubic reciprocity
Loewner's torus inequality
Hurwitz quaternion
Quadratic integer

Notes

^ Surányi, László (1997). Algebra. TYPOTEX. p. 73. and Szalay, Mihály (1991). Számelmélet. Tankönyvkiadó. p. 75. both call these numbers “Euler-egészek”, that is, Eulerian integers. The latter claims Euler worked with them in a proof.