Subgroups of Galois groups of finite fields

Question

According to the notion of Galois group, for $E=GF(2^n)$ as an extension of the field $F=GF(2)$, the Galois group $Gal(E/F)$ is a cyclic group of order $n$. Now my question is: for finding the subgroups of $Gal(E/F)$, do we just find the divisors of $2^n$? What are the generators of the subgroups?

Answer 1

For any prime $p$, the Galois group $\text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, hence has a unique subgroup of order $d$ for every $d$ dividing $n$ (not $p^n$). The entire Galois group is generated by the Frobenius automorphism, $\sigma : x \mapsto x^p$, so the $n/d^\text{th}$ power of the Frobenius, $\sigma^{n/d}$, will generate the subgroup of order $d$.

Answer 2

You're complicating the problem by keeping irrelevant details in your head. Finite fields and Galois theory are no longer relevant: the question you have asked is

What are the subgroups of the cyclic group on $n$ elements?