Simple Galois extension

Question

Let $\alpha$ be a algebraic real number over $\mathbb{Q}$, and let $\mathbb{Q}(\alpha)$ be a Galois over $\mathbb{Q}$.

Then, what can we say about the properties of its Galois group $\textrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})$?

For example:

1) $\textrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})$ is abelian.

2) There is a non-abelian group $\textrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})$ under the hypothesis.

I can't find the latter case. Can anyone help me? Thank you!