Eisenstein integer

In mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are complex numbers of the form

z = a + b\,\omega

where and a and b are integers and

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

is a complex cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane. 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(√−3). They also form a Euclidean domain.

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 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 or ±ω²}

These are just the Eisenstein integers with absolute value equal to one.

Eisenstein primes

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 and u where u is any of the six units.

It may be shown that an ordinary prime number (or rational prime) of the form $x^2 - xy + y^2$ may be factored into $(x + \omega y)(x + \omega^2 y)$ and is therefore not prime in the Eisenstein integers. Also, a number of the form x² − xy + y² is rational prime iff x + ωy is an Eisenstein prime.