Let ω ≠ 1 be a cube root of unity. Then the minimum of the set { a + b ω + c ω 2 2 : a …

If $z$ is a complex number
Then $|z{|}^2=z\overline{z}$
Now,
${\left|a+b\omega +c{\omega }^2\right|}^2=\left(a+b\omega +c{\omega }^2\right)\left(\overline{a+b\omega +c{\omega }^2}\right)$
a+bω+cω22=a+bω+cω2a+bω-+cω-2
$\left(ifa\in R\Rightarrow \stackrel{-}{a}=a\right)$
$\left(\overline{z_1{z}_2}={\stackrel{-}{z}}_1{\stackrel{-}{z}}_2\right)$
ω-=ω2,ω-2=ω

a+bω+cω22=a+bω+cω2a+bω2+cω
=a2+b2+c2-ab-bc-ca
$\left(\text{∵\hspace{0.17em}\hspace{0.17em}}{\omega }^3=1,1+\omega +{\omega }^2=0\right)$
|a+bω+cω2|2=12[a-b2+b-c2+c-a2]
Now for this to be minimum
Take a=1, b=2, c=3, as a, b, c are distinct and non-zero integers
Minimum of ${\left|a+b\omega +c{\omega }^2\right|}^2=3$