The common roots of the equations x^{12} - 1 | vedclass

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(C) Given equations are $x^{12} - 1 = 0$ and $x^4 + x^2 + 1 = 0$. Factorizing the first equation: $x^{12} - 1 = (x^6 - 1)(x^6 + 1) = (x^2 - 1)(x^4 + x

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$\pm \omega, \pm \omega^2$

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(C) Given equations are $x^{12} - 1 = 0$ and $x^4 + x^2 + 1 = 0$. Factorizing the first equation: $x^{12} - 1 = (x^6 - 1)(x^6 + 1) = (x^2 - 1)(x^4 + x^2 + 1)(x^6 + 1)$. Since $x^4 + x^2 + 1$ is a factor of $x^{12} - 1$,all roots of $x^4 + x^2 + 1 = 0$ are common roots. Solving $x^4 + x^2 + 1 = 0$: Let $y = x^2$,then $y^2 + y + 1 = 0$. The roots are $y = \omega$ and $y = \omega^2$,where $\omega = \frac{-1 + i\sqrt{3}}{2}$. Thus,$x^2 = \omega$ or $x^2 = \omega^2$. Taking square roots,$x = \pm \sqrt{\omega} = \pm \omega^2$ and $x = \pm \sqrt{\omega^2} = \pm \omega$. Therefore,the common roots are $\pm \omega, \pm \omega^2$.

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

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