Let $p ≠ 2$ be a prime number, and $\omega = e^{2πi/p}$. I now want to find the minimal polynomial of $\omega$ over the field $\mathbb{Q}[\omega + \omega^{-1}]$.
I must admit that I don't really know how to get started with this one. If this was over $\mathbb{Q}$, then I would probably choose a polynomial of which I know that it has $\omega$ as a root (I think $x^p - 1$ would be a fitting one to start with), and then try to split off factors, and see how far I can go with that.
But I don't really know how to approach this question when trying to find the minimal polynomial over a field like $\mathbb{Q}[\omega + \omega^{-1}]$. I was given the hint that I might be able to use the fact that $\omega + \omega^{-1} \in \mathbb{R}$ (of which I can easily see that it is true), but even that hint couldn't get me started so far.
Hint: In fact, we have $[\mathbb{Q}(\omega):\mathbb{Q}(\omega + \omega^{-1})]=2$. Consider here the monic polynomial $$f(x)=x^2-(\omega+\omega^{-1})x+1=0.$$ What are the roots of this polynomial ?