Eisenstein Integer - Eisenstein Primes

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 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.