Show that $\mathbb{Q}(\sqrt[3]{2})$ does not have non trivial automorphisms Since I started the abstract algebra course, I haven't had exercises associated with the existence or non-existence of automorphisms, and I haven’t found exercises similar to this either so I don't know how to proceed. I would greatly appreciate your help.

The Galois group of a field extension consists in the automorphisms that fix the base field and move the elements that generates the extension. If the extension is generated by a finite number of elements then the automorphisms characterized by the images of these elements, that are also roots of the minimal polynomial of these elements. Since the minimal polynomial of $\sqrt[3]{2}$ is $t^3-2$, which roots are $\sqrt[3]{2}, \sqrt[3]{2}w^2, \sqrt[3]{2}w^2$ where $w=e^{2\pi i/3}$, the unique roots in the extension is the first, so the only movement between roots is $\sqrt[3]{2}\to \sqrt[3]{2}$, that is the identity
