As in the title, I want to show that $\mathbb Q(i\sqrt[]{3} \sin(\frac{\pi}{5})-\cos(\frac{2\pi}{3}))$ is normal.
My idea was the following: Let $\alpha:=i\sqrt[] {3} \sin(\frac{\pi}{5})-\cos(\frac{2\pi}{3})$, then we can write $\alpha=(2\zeta_3-1)\cdot\Im(\zeta_{10})+\frac{1}{2}$ and now I guess we have $\mathbb Q(\alpha)$ is a subfield of $\mathbb Q(\zeta_{30})$, the $30$th cyclotomic field of $\mathbb Q$. And, since the galois group of $\mathbb Q(\zeta_{30})$ is abelian, we have that every subfield is normal over $\mathbb Q$. My problem: Is my assumption even true? And if so, how do I show it?