Let $\alpha$ be the root of $X^3-180$ in $\mathbb{C}/\mathbb{R}$ and let $L$ be the splitting field of $X^3-180$ over $\mathbb{Q}$. I've got to check out wether $\mathbb{Q}(\alpha)/\mathbb{Q}$ is normal and wether $L$ is a Galois extension over $\mathbb{Q}$.
Well, the polynomial is irreducible by Eisenstein and $L$ is a Galois extension as Char $(\mathbb{Q})=0$. I just don't know how to check out whether it is normal. In this case, $\mathbb{Q}(\alpha)$ has to include $^3\sqrt 180$, which isn't in $\mathbb{C}/\mathbb{R}$...