Find $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$, where $m|n$.
Is this proof correct?
Since the base field is finite, $\mathbb{F}_{p^m}\subset \mathbb{F}_{p^n}$ is a Galois extension. (Is there another easy way of seeing why $\mathbb{F}_{p^n}/\mathbb{F}_{p^m}$ is a Galois extension?) Therefore $|\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})| = [\mathbb{F}_{p^n}:\mathbb{F}_{p^m}]$. But $[\mathbb{F}_{p^n}:\mathbb{F}_{p}]= [\mathbb{F}_{p^n}:\mathbb{F}_{p^m}][\mathbb{F}_{p^n}:\mathbb{F}_{p}]$. Given the claim that $[\mathbb{F}_{p^k}:\mathbb{F}_{p}] = k$ for all $k$ we finally arrive at the answer $[\mathbb{F}_{p^n}:\mathbb{F}_{p^m}]=d$. The Galois group is cyclic (since the base field was finite) so $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m}) = \mathbb{Z}_d$