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:
 * $\cmod \alpha^2 = a^2 - ab + b^2$

where $\cmod {\, \cdot \,}$ denotes the modulus of a complex number.

Proof
We find that:

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

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

Also:

Therefore:
 * $\cmod \alpha^2 = a^2 + \paren {\omega + \overline \omega} a b + \omega \overline \omega b^2 = a^2 - a b + b^2$

as required.