Find the common roots of the equations $x^{12}-1=0$ and $x^4+x^2+1=0$

Question

OPTIONS

A) $\pm \omega$

B) $\pm \omega^2$

C) $\pm \omega, \pm \omega^2$

A very easy question obviously, but I have a few doubts.

From the second equation, it is obvious that $x^2$ is imaginary cube root of 1, generally represented by $\omega$ and $\omega^2$

So $$x=\pm \sqrt {\omega}, ~ \pm \omega$$

Which also works for the first equation

So the answer should be A),but instead it’s C). Why is $\pm \omega^2$ allowed?

Answer 1

As $w^3=1, w=w^4, \sqrt w=\pm w^2$