<s>[INST] Minimal polynomial- fields Let $\zeta$ = $\cos(\frac{2\pi}{7} ) + i\sin(\frac{2\pi}{7} )$ , let $\alpha = \zeta +\zeta^{-1} $  note that $\zeta^{-1} =\zeta^6 $
I try to find the minimal polynomial of $\alpha$ over $\mathbb Q$. 
I only managed to show that the degree of the minimal polynomial is 3. 
My attempt so far:
$\alpha^3 = \zeta^3+ 3\zeta^{-1}\zeta^2+3\zeta\zeta^{-2}+\zeta^{18}  = \zeta^3+\zeta^4+3\alpha            $
And I don't know how to continue, Thank you for your help 
 [/INST] Now add the equation
$$
α^4=ζ^4+ζ^{-4}+4(ζ^2+ζ^{-2})+6=ζ^4+ζ^3+4α^2-2
$$
to your consideration to find that
$$
α^4-α^3-4α^2+3α+2=0
$$
This has a root at $2=1+1^{-1}$, the remaining factor is
$$
α^3 + α^2 - 2α - 1=0
$$

You could of course also start at
$$
0=\frac{ζ^7-1}{ζ-1}=1+ζ+ζ^2+ζ^3+ζ^4+ζ^5+ζ^6=1+α+(α^2-2)+(α^3-3α)
$$
</s>