# Minimal polynomial of the form $\zeta_p+\frac{1}{\zeta_p}+\zeta_q+\frac{1}{\zeta_q }$?

We can calculate the minimal polynomial of $2cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over Q as x^3+x^-2x-1 and simlary for $2cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$.

Now my question is : Is there any way to calculate the minimal polynomial of $\zeta_7+\frac{1}{\zeta_7}+\zeta_5+\frac{1}{\zeta_5 }$? Thanks in advance

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The key is to use resultants. If $P$ is the minimal polynomial of $x$, and $Q$ is the minimal polynomial of $y$, then the resultant of $P(x)$ and $Q(z-x)$ as a polynomial in $z$ vanishes when $z=x+y$. Factoring gives you the minimal polynomial of $x+y$.