Section 8.2 Eisenstein integer
\begin{equation}
x^2+x+1,\quad\text{i.e.,}\quad\omega^2+\omega+1=0.\tag{8.2.1}
\end{equation}
Checkpoint 8.2.2.
Lemma 8.2.3.
The map
\begin{equation*}
N\colon\Z[\omega]\to\Z\quad\text{given by}\quad a+b\omega\mapsto a^2+b^2-ab
\end{equation*}
is multiplicative and non-negative. Furthermore, \(N(a+b\omega)=0\) if and only if \(a=b=0\text{.}\)Proof.
\begin{equation*}
a^2+b^2-ab=\left(a-\frac{b}{2}\right)^2+\frac{3}{4}b^2.
\end{equation*}
This observation may be used to get a proof of the result.Lemma 8.2.4. (Units in Eisenstein integers).
An element \(a+b\omega\in\Z[\omega]\) is a unit if and only if \(N(a+b\omega)=1\text{.}\) Only units in \(\Z[\omega]\) are \(\pm 1,\pm\omega,\pm\omega^2\text{.}\)Proof.
\begin{equation*}
(a,b)=\{(\pm 1,0),(0,\pm 1),(-1,-1),(1,1)\}
\end{equation*}
and corresponding Eisenstein integers are
\begin{equation*}
\{\pm 1,\pm\omega,-1-\omega,1+\omega\}
\end{equation*}
Every element in the above set is a unit (see equation (8.2.1)). Furthermore, \(\pm(1+\omega)=\mp\omega^2\) by equation (8.2.1).
