Complex numbers with integers

$let~\omega~be~w$ ,just to avoid typing.

The expression is $|a+bw+cw^{2}|$ as $\frac{1}{w}=w^{2}$

$|a+bw+c{ w }^{ 2 }|^{ 2 }=(a+bw+c{ w }^{ 2 })(\overline { a+bw+c{ w }^{ 2 } } )$

$|a+bw+c{ w }^{ 2 }|^{ 2 }=a^{2}+b^{2}+c^{2}-(ab+bc+ca)$

using $w^{3}=1~and~w+w^{2}=-1$ we get the above expression

$\large{|a+bw+c{ w }^{ 2 }|^{ 2 }=\frac{1}{2}((a-b)^{2}+(b-c)^{2}+(c-a)^{2})}$

since $a,b,c$ are distinct integers $min(a-b)=1~and~min(b-c)=1~and~min(c-a)=2$

or u can put $a=1,~b=2~,c=3$

$\large{min|a+bw+c{ w }^{ 2 }|^{ 2 }=\frac{1}{2}(1+1+4)}$

$\huge{min|a+bw+c{ w }^{ 2 }|=\sqrt{3}}$

cheers!!