<s>[INST] 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.
 [/INST] Like factor $z^7-1$ into linear and quadratic factors and prove that $ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$,
we can prove $a^r+a^{7-r}=a^r+a^{-r}=2\cos\dfrac{2r\pi}7$ for $r=1,2,3$
Now if $7x=2r\pi,$
$\cos4x=\cos(2r\pi-3x)=\cos3x\  \ \ \ (1) $
So, the equation whose roots are $\cos\dfrac{2r\pi}7$ for $r=0,1,2,3$
$$8c^4-8c^2+1=4c^3-3c$$
But $(1)\implies4x=2m\pi\pm3x$ where $m$ is any integer
$4x=2m\pi+3x\implies x=2m\pi,\cos x=1$
So, the equation whose roots are $\cos\dfrac{2r\pi}7$ for $r=1,2,3$
$$\dfrac{8c^4-4c^3-8c^2+3c+1}{c-1}=0$$
Can you take it from here? 
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