TITLE: Minimal polynomial of $\omega:=\zeta_7+\overline{\zeta_7}$
QUESTION [5 upvotes]: Let $\omega:=\zeta_7+\overline{\zeta_7}$, where $\zeta_7$ is a primitive $7$th root of $1$. I want to find the minimal polynomial of $\omega$ over $\mathbb{Q}$. 
I've found
$$\omega=\zeta_7+\overline{\zeta_7}\quad;\quad\omega^2=\zeta_7^2+\overline{\zeta_7}+2\quad;\quad\omega^3=\zeta_7^3+\overline{\zeta_7}^3+3\omega$$
Now how can I find the minimal polynomial?

REPLY [2 votes]: You seek to solve
$$ a + b \omega + c \omega^2 + d \omega^3 = 0$$
for $(a,b,c,d)$. That looks like a linear algebra problem.
The general purpose method is:

Choose a basis for the 6-dimensional rational vector space $\mathbb{Q}(\zeta_7)$.
Find the coefficients of $\omega^i$  with respect to this basis
Arrange them into a suitable matrix equation
Solve.