A Field Extension of a Cyclic Galois Group is Galois

Question

Let $F \subseteq E $ be extension of fields. If $Gal(E/F)$ is a cyclic group, does it imply that the extension $E/F$ is a Galois extension? If not, any example?

Answer 1

It depends. First one should note that usually writing $\text{Gal}(E/F)$ is reserved for when the extension is already known to be Galois, and we write $\text{Aut}(E/F)$ otherwise, but this is not that important. Now let $\mathbb{F}_p(t^p)[x]/(x^p - t^p) = E$, and $\mathbb{F}_p(t^p) = F$, this has trivial automorphism group and is not Galois as it is not separable. In particular now one can take the compositum $E = \mathbb{F}_q(t^p)[x]/(x^p - t^p)$, $F = \mathbb{F}_p(t^p)$ which should have galois group $\mathbb{Z}/n\mathbb{Z}$ where $q = p^n$ but not be a Galois extension.

In general there are not any conditions I am aware of that one can put on the automorphism group of a field extension that is not Galois that ensure that it is Galois.