Field extension if element fixed by only identity

Question

Suppose that $E$ is a Galois extension of $F$ and that $α \in E$ is left fixed by only the identity in $\text{Gal}(E/F)$. Prove that $E = F (α)$.

Please suggest how I should proceed. Thanks!

Answer 1

Since the extension is Galois, the fundamental theorem of the Galois theory says that there exists a bijection between subfields of $E$ and subgroups of $Gal(E/F)$ by the correspondence which sends $H\subset Gal(E/F)$ to $E^H$.

The subgroup $H$ such that $F(\alpha)=E^H$ is the trivial group $\{1\}$ here thus $E=F(\alpha)$.