Eisenstein integer

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

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

z = a + b ω

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.

1 Properties
2 Eisenstein primes
3 Euclidean domain
4 See also
5 External links

[edit] 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 - (2 a - b) z + (a 2 - a b + b 2).

In particular, ω satisfies the equation

ω 2 + ω + 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.

[edit] 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 the an ordinary prime number (or rational prime) which is 3 or congruent to 1 mod 3 is of the form $x 2 - x y + y 2$ for some integers x,y and may be therefore factored into $(x + ω y)(x + ω 2 y)$ 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. Also, a number of the form x² − xy + y² is rational prime iff x + ωy is an Eisenstein prime.