Galois group of extensions over quasi-finite fields A field $F$ is called quasi-finite if it is perfect and for every $n>0$, there is exactly one extension of degree $n$. If $F$ is a quasi-finite field, and $E/F$ is a finite extension, show that$$\text{Gal}(E/F)$$
is a cyclic group.
I know that if $F$ is finite, then $F$ is quasi-finite, and if $\text{car}(F) =p $, then the Fröbenius automorphism generates $\text{Gal}(E/F)$.
Another example (taken from Wikipedia) is $K$, the ring of Laurent series over $\mathbb C$, when considering the unique extension $K_n = \mathbb C((T^{1/n}))$. Then a generator of $\text{Gal}(K_n/K) $ is given by:
$$F_n(T^{1/n}) = e^{\frac{2\pi i}{n}}T^{1/n}$$
Is it possible to find a generator for  $\text{Gal}(E/F)$ explicitly? Also, the example from Wikipedia is not entirely clear to me, since $K$ is not a field (How can it be quasi-finite?). If the generator cannot be found, what is a good way to show this? Thanks.
 A: Since $E/F$ is finite then there exists the normal closure of $E/F$, say $K/F$. Then $K/F$ is the splitting field of some polynomial $f(x) \in F[x]$. Since $F$ is perfect then $f(x)$ is separable and hence $K/F$ is separable. So, $K/F$ is a Galois extension. By the Fundamental Theorem of Galois Theory, $\mathrm{Gal}(K/F)$ has a subgroup for each intermediate field of $K/L$
Now, each intermediate field of $K/L$ is a finite extension $F$.Therefore, since $F$ is quasi-finite, the intermediate fields of $K/F$ are extensions of $F$ of distinct degrees, because each $n \in \mathbb{N}^*$, $F$ can have one extension of degree $n$, at most. Hence, the subgroups of $\mathrm{Gal}(K/F)$ have different order. 
If $|\mathrm{Gal}(K/F)|=n$, for each $d \in \mathbb{N}$ such that $d \mid n$, it will occur that $\mathrm{Gal}(E/F)$ has at most one subgroup of order $d$. Therefore $\mathrm{Gal}(K/F)$ is cyclic. 
Finally, if $\sigma \in \mathrm{Gal}(E/F)$, since $K/F$ is normal, $\sigma$ extends to an element of $K/F$. Hence $\mathrm{Gal}(E/F)$ corresponds to a subgroup of $\mathrm{Gal}(K/F)$, i.e. there is a monomorphism $\varphi:\mathrm{Gal}(K/F) \to S$, where $S$ is a subgroup of $\mathrm{Gal}(K/F)$. Since every subgroup of a finite and cyclic group is cyclic, we conclude $\mathrm{Gal}(E/F)$ is cyclic. 
