\begin{definition}[Definition:Eisenstein Integer]
An '''Eisenstein integer''' is a complex number of the form
:$a + b \omega$
where $a$ and $b$ are both integers and:
:$\omega = e^{2 \pi i / 3} = \dfrac 1 2 \paren {i \sqrt 3 - 1}$
that is, the (complex) cube roots of unity.
The set of all '''Eisenstein integers''' can be denoted $\Z \sqbrk \omega$:
:$\Z \sqbrk \omega = \set {a + b \omega: a, b \in \Z}$
\end{definition}