Hi all I am in higher level mathematics and I am taking the IB. We started doing problems associated with nth root of unity. I understand how to find the roots of for example:
$$Z^3 - 1 =0$$
and also understand how the roots will equal 0 when added up. Furthermore my teacher showed me how when:
$Z_2$ (when $k = 1$) $= w$, that means $(Z_2)^2 = w^2 = Z_3$ (when $k = 2$).
He then made us note $Z^3-1 = (Z-1)(Z^2+Z+1)$
And then made us state: if $w$ is the root with the smallest positive argument, that means $Z^2+Z+1=0$ which means $w^2+w+1= 0$ since $w-1$ cannot $= 0$.
The above statement I do not understand, can someone explain this to me since I need it to simplify:
can someone please explain this to me, I will be so grateful thanks so much.