If a is a non real root of $x^7 = 1$, find the equation whose roots are $ a + a^6 , a^2 + a^5, a^3 + a^4$ If a is a non real root of $ x^7 = 1$, find the equation whose roots are $a + a^6 , a^2 + a^5, a^3 + a^4$. This is one of the questions I have encountered while preparing for pre rmo. I feel the question requires the concept of the nth roots of unity and de moivre's theorem. But i actually couldnt work it out. Any help will be appreciated. Thanks in advance.

WLOG let $a=e^{\frac{2i\pi}{7}}$. Then expand 
$$(x-(a+a^6))(x-(a^2+a^5))(x-(a^3+a^4))$$
using the facts that $a^7=1$ and $a^6+a^5+a^4+a^3+a^2+a+1=0$. 
