Minimizing $|a+bw+cw^2|$ where $a,b,c\in\mathbb Z$ and $w = \zeta_3$. 
If $a,b,c\in\mathbb Z$ are not all equal and $w$ is a cube root of unity $(w\neq 1$), then the minimum value of:
  $$|a+bw+cw^2|$$
  is what?

I'm pretty stuck with the above problem.  Could someone help me out?

Using $|z|^2 = z\bar{z}$
and Using $\displaystyle \bar{\omega} = \omega^2$ and $\bar{\omega^2} = \omega$ and $\omega^3 = 1$ and $1+\omega+\omega^2 = 0$
So $$|a+b\omega+c\omega^2|^2 = (a+b\omega+c\omega^2)\cdot \bar{(a+b\omega+c\bar{\omega^2})} = (a+b\omega+c\omega^2)(a+b\omega^2+c\omega)$$
So we get $$|a+b\omega+c\omega^2|^2=a^2+b^2+c^2+ab(\omega+\omega^2)+bc(\omega+\omega^2)+ca(\omega+\omega^2)$$
So $$|a+b\omega+c\omega^2|^2=a^2+b^2+c^2-ab-bc-ca = \frac{1}{2}\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$$
Now Given $a,b,c$ not all are equal integers. So for $\min$ of right side expression 
Let $a=n\;,b=n\;,c=n+1\;,$ Where $n\in \mathbb{Z}$
So $$\min|a+b\omega+c\omega^2|^2 = \frac{1}{2}\left[1^2+1^2\right] = 1$$
So $$\min |a+b\omega+c\omega^2| = 1$$
