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 + isqrt 3) = e^{2pi i/3}

is a complex cube root of unity.

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 + b ω 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.

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 divides for ordinary integers. Therefore we may also extend the notion of primality; an Eisenstein integer x is said to be an Eisenstein prime if its only divisors are :pm x, pmomega x, pmomega^2 x, pm1, pmomega, pmomega^2 (except that we do not consider ±1, ±ω or ±ω2 themselves to be Eisenstein primes — they are unit (ring theory)s in the ring of integers).

= Relation to primes of the form x 2 − xy + y 2 =

It may be shown that a 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 x 2 − xy + y 2 is prime iff x + ωy is an Eisenstein prime.

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