Norm of Eisenstein Integer

Theorem
Let $\alpha$ be an Eisenstein integer.

That is, $\alpha = a + b \omega$ for some $a,b \in \Z$, where $\omega = e^{2\pi i /3}$.

Then:
 * $\left|{\alpha}\right|^2 = a^2 - ab + b^2$

where $\left|{\cdot}\right|$ denotes the modulus of a complex number.

Proof
We find that:

By the definition of the polar form of a complex number:
 * $\omega = \exp \left({\dfrac {2 \pi i} 3}\right) = \cos \left({\dfrac {2 \pi} 3}\right) + i \sin \left({\dfrac {2 \pi} 3}\right) = -\dfrac 1 2 + i \dfrac {\sqrt 3} 2$

Thus by Sum of Complex Number with Conjugate:
 * $\omega + \overline \omega = 2 \cdot \left({ - \dfrac 1 2 }\right) = -1$

Also:

Therefore:
 * $\left\vert{\alpha}\right\vert^2 = a^2 + \left({\omega + \overline \omega}\right) a b + \omega \overline \omega b^2 = a^2 - a b + b^2$

as required.