Let ω ≠ 1 be a cube root of unity. Then the minimum of the set |a +...

If $z$ is a complex number
Then $| z{|}^{2} = z\overset{¯}{z}$
Now,
${\left|a + b\omega + c{\omega}^{2}\right|}$^{2} = \left(a + b\omega + c{\omega}^{2}\right)\left(\overset{¯}{a + b\omega + c{\omega}^{2}}\right)
${\left|a + b\omega + c{\omega}^{2}\right|}$^{2} = \left(a + b\omega + c{\omega}^{2}\right)\left(a + b\overset{-}{\omega} + c{\overset{-}{\omega}}^{2}\right)
\left(ifa \in R ⇒ \overset{-}{a} = a\right)
\left(\overset{¯}{{z}_{1}{z}_{2}} = \overset{-}{z}1\overset{-}{z}2\right)
\left(\overset{-}{\omega} = {\omega}^{2}, {\overset{-}{\omega}}^{2} = \omega\right)

${\left|a + b\omega + c{\omega}^{2}\right|}$^{2} = \left(a + b\omega + c{\omega}^{2}\right)\left(a + b{\omega}^{2} + c\omega\right)
$= \left({a}^{2} + {b}^{2} + {c}^{2} - ab - bc - ca\right)$
\left(\text{∵}{\omega}^{3} = 1, 1 + \omega + {\omega}^{2} = 0\right)$<br />$| a + b\omega + c{\omega}^{2}{|}^{2} = \dfrac{1}{2} [ {\left(a - b\right)}^{2} + {\left(b - c\right)}^{2} + {\left(c - a\right)}^{2} ]$<br />Now for this to be minimum<br />Take$a = 1, b = 2, c = 3,$as$a, b, c$are distinct and non-zero integers<br />Minimum of${\left|a + b\omega + c{\omega}^{2}\right|}^{2} = 3$