Unique subfield of order $p^d$ in a field with $p^n$ elements

Let \(F=\mathbb F_{p^n}\). The Frobenius automorphism \[ \varphi(x)=x^p \] generates \[ \operatorname{Gal}(F/\mathbb F_p)\cong C_n. \] If \(d\mid n\), then \(\langle\varphi^d\rangle\) has order \(n/d\), so its fixed field has degree \(d\) over \(\mathbb F_p\) and therefore has \(p^d\) elements. Explicitly, that fixed field is \[ F_d=\{x\in F:x^{p^d}=x\}. \] Thus a subfield of order \(p^d\) exists for every \(d\mid n\).

It is unique. If \(K\subseteq F\) has \(p^d\) elements, then every \(x\in K\) satisfies \(x^{p^d}=x\), so \[ K\subseteq F_d. \] Both fields have \(p^d\) elements, hence \(K=F_d\).

Conversely, if \(K\subseteq\mathbb F_{p^n}\) has \(p^d\) elements, then \[ p^n=|K|^{[F:K]}=p^{d[F:K]}, \] so \(d\mid n\). Therefore the subfields of \(\mathbb F_{p^n}\) are exactly the unique fields \(\mathbb F_{p^d}\) for divisors \(d\) of \(n\).