Size of $\mathrm{Gal}(\mathbb{Q}(\cos(2\pi/7))/\mathbb{Q})$

Question

My objective is to prove that $|\mathrm{Gal}(\mathbb{Q}(\cos(2\pi/7))/\mathbb{Q})|=3$. So far, I've been able to show that the degree of the field extension $\mathbb{Q}(\cos(2\pi/7))/\mathbb{Q}$ is $3$ using that $\mathbb{Q}(e^{2\pi/7})/\mathbb{Q}(\cos(2\pi/7))$ has degree 2 and $\mathbb{Q}(e^{2\pi/7})/\mathbb{Q}$ has degree 6. I don't know how to proceed from here. The standard way I know is to find the minimal polynomial of $\cos(2\pi/7)$ over the rationals and presenting the Galois group of the extension as a group of permutations of the roots of such polynomial contained in $\mathbb{Q}(\cos(2\pi/7))$. If I could somehow prove that all such roots are contained in $\mathbb{Q}(\cos(2\pi/7))$ it would be enough but I would like to avoid having to calculate the minimal polynomial. Thank you in advance.

Answer 1

The answer is almost contained in the answers of this question Galois group of $x^3 + x^2 - 2x - 1$.

The second answer shows that the minimal polynomial of $\cos({{2\pi}\over 7})$ is $x^3-x^2-2x-1$ and the first and second answer shows that the Galois group of this polynomial has order $3$.