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(A) The given equation is $\alpha^{2}+\alpha+1=0$. This is the characteristic equation for the cube roots of unity,where $\alpha$ is either $\omega$ o

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$\alpha$

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(A) The given equation is $\alpha^{2}+\alpha+1=0$. This is the characteristic equation for the cube roots of unity,where $\alpha$ is either $\omega$ or $\omega^{2}$. We know that $\omega^{3}=1$. For $\alpha=\omega$,we have $\alpha^{31} = \omega^{31} = (\omega^{3})^{10} \cdot \omega = 1^{10} \cdot \omega = \omega = \alpha$. For $\alpha=\omega^{2}$,we have $\alpha^{31} = (\omega^{2})^{31} = \omega^{62} = (\omega^{3})^{20} \cdot \omega^{2} = 1^{20} \cdot \omega^{2} = \omega^{2} = \alpha$. In both cases,$\alpha^{31} = \alpha$.

If $\alpha$ is a complex number satisfying the equation $\alpha^{2}+\alpha+1=0$,then $\alpha^{31}$ is equal to

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