[step: Every subfield of $\mathbb{F}_{p^n}$ arises as the fixed field of a subgroup of $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$] The extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois of degree $n$, with Galois group generated by the Frobenius automorphism $\varphi\colon x\mapsto x^p$. Since $\varphi$ has order $n$, we have $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p) = \langle\varphi\rangle \cong \mathbb{Z}/n\mathbb{Z}$. By the Fundamental Theorem of Galois Theory, intermediate fields of $\mathbb{F}_{p^n}/\mathbb{F}_p$ correspond bijectively to subgroups of $\mathbb{Z}/n\mathbb{Z}$. Because $\mathbb{Z}/n\mathbb{Z}$ is cyclic, its subgroups are exactly the cyclic subgroups $\langle \bar{d}\rangle$ of order $n/d$, one for each positive divisor $d$ of $n$. Translating back, the subgroups of $\langle\varphi\rangle$ are exactly $\langle\varphi^d\rangle$ for $d\mid n$, each of order $n/d$. [step: The fixed field of $\langle\varphi^d\rangle$ is $\mathbb{F}_{p^d}$] The fixed field of $\langle\varphi^d\rangle$ is \begin{align*} \mathbb{F}_{p^n}^{\langle\varphi^d\rangle} &= \{x \in \mathbb{F}_{p^n} : \varphi^d(x) = x\} \\ &= \{x \in \mathbb{F}_{p^n} : x^{p^d} = x\}. \end{align*} This is precisely the splitting field of $X^{p^d} - X$ inside $\mathbb{F}_{p^n}$, which is $\mathbb{F}_{p^d}$. (The polynomial $X^{p^d}-X$ has exactly $p^d$ roots in any field that contains them, and these roots form a subfield of order $p^d$; the inclusion $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ holds because $d\mid n$ implies $X^{p^d}-X$ divides $X^{p^n}-X$.) [guided: Confirm the correspondence is exhaustive] The subgroup $\langle\varphi^d\rangle$ has order $n/d$, so by the Fundamental Theorem its fixed field has degree $[\mathbb{F}_{p^n}:\mathbb{F}_{p^n}^{\langle\varphi^d\rangle}] = n/d$, giving $[\mathbb{F}_{p^n}^{\langle\varphi^d\rangle}:\mathbb{F}_p] = d$. This matches $|\mathbb{F}_{p^d}| = p^d$. Since we have exhausted all subgroups of $\mathbb{Z}/n\mathbb{Z}$, the subfields of $\mathbb{F}_{p^n}$ are exactly $\mathbb{F}_{p^d}$ for $d\mid n$. $\blacksquare$