The common roots of the equation ${\mathrm{x}}^{12}-1=0,{\mathrm{x}}^4+{\mathrm{x}}^2+1=0$ are

$\begin{array}{r}{\mathrm{x}}^{12}-1=0\Rightarrow {\left({\mathrm{x}}^6\right)}^2-1=\left({\mathrm{x}}^6-1\right)\left({\mathrm{x}}^6+1\right)\\ =\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^4+{\mathrm{x}}^2+1\right)\left({\mathrm{x}}^6+1\right)\\ \therefore common roots of {\mathrm{x}}^{12}-1=0 and {\mathrm{x}}^4+{\mathrm{x}}^2+1=0\\ are same as roots of x^4+{\mathrm{x}}^2+1=0\\ x^2=\omega or {\omega }^2\Rightarrow x=\pm \omega ,\pm {\omega }^2\end{array}$