Eisenstein integers; finding a condition to be a unit using a multiplicative function.

Question

Context $1$: Let $E = \Bbb Z [w] = \{m + nw \in \Bbb C \hspace{.1cm}| \hspace{.1cm} m,n \in \Bbb Z\}$ the domain of the Eisenstein integers. Doing some work, we know that $\omega^2 = \bar{\omega} $. Also, $\omega^3 = 1 \Leftrightarrow \omega \bar{\omega} = 1$. Furthermore, $\omega^2 + \omega = -1$.

Showing that $E$ is a domain. Since $E \subset \Bbb C$ and $\Bbb C$ is ring, then $E$ is a subring of $\Bbb C$. This also implies that $E$ is a subdomain of $\Bbb C$ and thus it is also a domain.

Context $2$: Define the function $\eta: E \rightarrow \Bbb N_0$ such that $z \rightarrow |z|^2$. From this we can easily see that this function is multiplicative since the norm is, and it satisfies $\eta(z) = 0 \Leftrightarrow z = 0$. Furthermore, $\eta(m+n\omega) = m^2 + n^2 - mn$. This is an euclidian evalutation and so $E$ is an euclidian domain. Note that $\eta(m+n\omega) = \frac{1}{4}((2m-n)^2+3n^2)$.

Question. Show that $m+n\omega$ is a unit iff $\eta(m+nw) = 1$.

My attempt. Suppose that $u = m + n\omega$ is a unit in $E$. Then, we have that \begin{equation*} 1 = \eta(1) = \eta(u u^{-1}) = \eta(u) \eta(u^{-1}) \end{equation*} Obviously, both $\eta(u)$ and $\eta(u^{-1})$ must be in $\Bbb N_0$ and so the only option we have is that $\eta(u) = 1 = \eta(u^{-1})$. This proves that $\eta(u) = \eta(m+n\omega) = 1$.

Now, we need to prove that $\eta(u) = \eta(m+nw) = 1 \Rightarrow u$ is an unit in $E$. I don't know how to proceed to prove this implication.

Thanks for any help in advance.

Answer 1

$\eta(a+b\omega)=(a+b\omega)(a+b\bar\omega)$ where $a,b$ are integers. So if $\eta(a+b\omega)=1$ then $(a+b\omega)(a+b\bar\omega)=1$ and $a+b\omega$ is a unit.

Answer 2

We already know that an element $u$ in the ring of integers $\mathcal{O}_K$ of a number field $K$ is a unit if and only if its norm equals $\pm 1$, see for example this duplicate:

Units in ring of integers are exactly those with norm {-1,1}

For $K=\Bbb Q(\omega)$, the norm of an element $a+b\omega$ in $\mathcal{O}_K=\Bbb Z[\omega]$ cannot be negative, because of $$N(a+b\omega)=a^2-ab+b^2 = \left( a - \frac{1}{2}b\right)^2 + \frac{3}{4}b^2,$$ as you have stated in your context 2 text (with $\eta(n+m\omega)$). So we are done.