How would one find the minimal polynomial of $e^{2\pi i/5}$ over $\mathbb{Q}(\cos(2{\pi}/5))$?
The $\mathbb{Q}(\cos(2{\pi}/5))$ is what is confusing me the most. I know that $\mathbb{Q}(\cos(2\pi/5))$ is an extension of $\mathbb{Q}$ where we have polynomials with rational numbers and $\cos(2\pi/5)$ as coefficients, but I don't know how to combine with finding the minimal polynomial.
I tried setting $x=e^{2\pi i/5}$ and I know that $x^5=1$. Then I could do $$x^5-1=0 \rightarrow x^5-1=(x-1)(x^4+x^3+x^2+x+1)$$ but after that I don't know how to proceed.