minimal polynomial of $7^{th}$ root of unity 
Let $\alpha=\zeta_7+\zeta_7^2+\zeta_7^4$. What's the minimal polynomial of $\zeta_7$ over $\mathbb{Q}(\alpha).$

I have managed to show that $[\mathbb{Q}(\alpha):\mathbb{Q}]=2$ and $[\mathbb{Q}(\zeta):\mathbb{Q}]=6$. By tower rule, the minimal polynomial has degree 3. As long as we find a polynomial with degree 3 such that $\zeta$ is a root, then we are done. But I got stuck in finding such polynomial. Any hint is appreciated.

Put $\beta=\zeta^{-1}+\zeta^{-2}+\zeta^{-4}$ and note $\beta=-1-\alpha\in\mathbb{Q}(\alpha)$. Now form the cubic with roots $\zeta, \zeta^2,\zeta^4$. The sum of the roots is $\alpha$, the sum of the roots two at a time is $\beta$, the product of the roots is 1.
