Eisenstein Integers form Integral Domain

Theorem
The ring of Eisenstein integers $\left({\Z \left[{\omega}\right], +, \times}\right)$ is an integral domain.

Proof
By Ring of Eisenstein Integers we know that $\Z \left[{\omega}\right]$ is a subring of the complex numbers $\C$.

Let $1_\C$ be the unity of $\C$.

Let $1_\omega$ be the unity of $\Z \left[{\omega}\right]$.

By the subdomain test it suffices to show that $1_\C = 1_\omega$.

By Unity of Ring is Unique it suffices to show that $1_\C$ is a unity of $\Z \left[{\omega}\right]$.

First we note that
 * $\Z \left[{\omega}\right] = \left\{{ a + b\omega : a,b \in \Z }\right\}$

In particular, $1_\C \in \Z \left[{\omega}\right]$.

Moreover by definition $\Z \left[{\omega}\right]$ inherits it's multiplication from $\C$.

For any $\alpha \in \Z \left[{\omega}\right]$ we have:
 * $1_\C \alpha = \alpha 1_\C = \alpha$

in $\C$.

Therefore this identity holds in $\Z \left[{\omega}\right]$ as well.