Eisenstein integer

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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 + i\sqrt 3) = e^]

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

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.

Euclidean domain

The ring of Eisenstein integers forms a Euclidean domain whose norm v is

[ v(a + \omega b) = a^2 - a b + b^2. ]

This can be derived by embedding the Eisenstein integers in the complex numbers: since

[ v(a + i b) = a^2 + b^2 ]

and since

[ a + \omega b = \left( a - b\right) + i \over 2} b]

it follows that

[ v(a + \omega b) = \left( a - b\right)^2 + b^2 ]

::[ = a^2 - a b + b^2 + b^2 = a^2 - a b + b^2].

See also

• Eisenstein prime
• Gaussian integer
• Kummer ring

External links

• [Eisenstein Integer--from MathWorld]

 

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