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 we have:
 * $\omega = \exp\left({ \dfrac{2 \pi i} 3 }\right) = \cos\left({\dfrac{2 \pi} 3}\right) + i \sin\left({\dfrac{2 \pi} 3}\right) = -\dfrac 1 2 + i\dfrac {\sqrt{3}} 2$

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

We also find that:

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

as required.