Calculate $\mathbb{Q(\alpha)} / \mathbb{Q(\alpha^3)}$ where $\alpha$ is transcendental over $\mathbb{Q}$

Question

I am doing a long problem of Galois theory and got stuck here:

Let $\alpha\in\mathbb{R} ,\alpha\notin \mathbb{Q}$ be transcendental (non-algebraic) over $\mathbb{Q}$, calculate [$\mathbb{Q(\alpha)}:\mathbb{Q(\alpha^3)}$]. Is $\mathbb{Q(\alpha)} / \mathbb{Q(\alpha^3)}$ a Galois extension?

I don't know if this can help, but in a previous section of the problem I proved that $\mathbb{Q(\alpha)} / \mathbb{Q(\alpha^3)}$ is algebraic and separable. (In this section $\alpha$ was not said to be transcendental over $\mathbb{Q}$)