Eisenstein Integer - Properties

Properties

The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω) — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial

In particular, ω satisfies the equation

The product of two Eisenstein integers and is given explicitly by

The norm of an Eisenstein integer is just the square of its absolute value and is given by

Thus the norm of an Eisenstein integer is always an ordinary (rational) integer. Since

the norm of a nonzero Eisenstein integer is positive.

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, ±ω2}

These are just the Eisenstein integers of norm one.