I have problem with these 2 exercices:

1) Multiply out and simplify $(a+bw)(a+bw^2)$ where $ω=e^{2πi/3}$ I only know that$(a+bw)(a+bw^2)= a^2+abw^2+abw+b^2$ and I don't know what to do next

2) If $ω$ is a complex third root of unity and $x$ and $y$ are real numbers prove that: $(a) 1+ω+ω^2=0$

$(b) (ωx+ω^2y)(ω^2x+ωy)=x^2-xy +y^2 $

and I have no idea even how to start

Thank you in advance for any help!


We know that $\omega$ is a complex root of the equation $x^3-1=0$. This can be reduced to


If we plug in $\omega$ into this, you'll get $(\omega-1)(\omega^2+\omega+1)=0$

Now we know that $\omega \ne 1$. You can complete the rest yourself.


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