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.

Think of the roots as $a+a^{-1}$, $a^2+a^{-2}$ and $a^3+a^{-3}$. Then $a$ satisfies $a^6+a^5+a^4+a^3+a^2+a+1=0$ or equivalently,
$a^3+a^2+a+1+a^{-1}+a^{-2}+a^{-3}=0$. Can you write the expression
$x^3+x^2+x+1+x^{-1}+x^{-2}+x^{-3}$ as a linear combination of $(x+x^{-1})^3$, $(x+x^{-1})^2$, $x+x^{-1}$ and $1$? Putting in $a$ for $x$
would then give you an equation satisfied by $a+a^{-1}$.
