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