Complex Numbers

goals
Show that $(x \, + \, y)^2 \, + \, (x\omega \, + \, y\omega^2)^2 \, + \, (x\omega^2 \, + \, y\omega)^2 = 6xy.$

Show that $(x \, + \, y)^2 \, + \, (x\omega \, + \, y\omega^2)^2 \, + \, (x\omega^2 \, + \, y\omega)^2 = 6xy.$

If $a = \cos \dfrac {2\pi}{7} + i\sin \dfrac {2\pi}{7}$, then the quadratic equation whose roots are $\alpha = a + a^{2} + a^{4}$ and $\beta = a^{3} + a^{5} + a^{6}$, is
If  $a+ib=\dfrac {x+iy}{x-iy}$ prove that $a^2+b^2 =1$ and $\dfrac {b}{a}=\dfrac {2xy}{x^2-y^2}$
In the multiplicative group of $n$th roots of unity, the inverse of ${ \omega }^{ k }$ is $(k< n)$: