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(D) The roots of the equation $x^2+x+1=0$ are the complex cube roots of unity, $\omega$ and $\omega^2$.Let $\alpha = \omega$ and $\beta = \omega^2$.We

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$x^2+x+1=0$

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(D) The roots of the equation $x^2+x+1=0$ are the complex cube roots of unity, $\omega$ and $\omega^2$.Let $\alpha = \omega$ and $\beta = \omega^2$.We need to find the equation whose roots are $\alpha^{19}$ and $\beta^7$.Since $\omega^3 = 1$, we have $\alpha^{19} = \omega^{19} = (\omega^3)^6 \cdot \omega = 1^6 \cdot \omega = \omega$.Similarly, $\beta^7 = (\omega^2)^7 = \omega^{14} = (\omega^3)^4 \cdot \omega^2 = 1^4 \cdot \omega^2 = \omega^2$.The new roots are $\omega$ and $\omega^2$, which are the same as the original roots.Therefore, the required equation is $x^2+x+1=0$.

If $\alpha, \beta$ are the roots of the quadratic equation $x^2+x+1=0$, then the equation whose roots are $\alpha^{19}, \beta^7$ is

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