The Galois description of the subfields of a finite field and the elementary divisibility criterion agree

Two statements in this library describe the subfields of a finite field, and they describe the same objects.

The subfields of ${\mathbb{F}}_{p^n}$ are the unique fields ${\mathbb{F}}_{p^d}$ for positive divisors $d$ of $n$ fixes a finite field $F$ of order $p^m$ with $p$ its characteristic and says that for each positive divisor $e$ of $m$ the set $\{a\in F:a^{p^e}=a\}$ is the unique subfield of $F$ of order $p^e$, and that these are all of the subfields of $F$. Its index set is therefore the divisors of $m$, and its base point is the prime field.

The intermediate fields of ${\mathbb{F}}_{q^n}\mathrm{/}{\mathbb{F}}_q$ are the ${\mathbb{F}}_{q^d}$, one for each positive divisor $d$ of $n$ fixes a base field ${\mathbb{F}}_q$ inside $F$ and says that the intermediate fields of $F\mathrm{/}{\mathbb{F}}_q$ are the $\{x\in F:x^{q^d}=x\}$ for the positive divisors $d$ of $n=[F:{\mathbb{F}}_q]$. Its index set is therefore the divisors of $n$, and its base point is ${\mathbb{F}}_q$.

The dictionary. Write $q=p^k$, so that $m=kn$. For a positive divisor $d$ of $n$ the two prescriptions produce literally the same set,

$\{x\in F:x^{q^d}=x\}=\{x\in F:x^{p^{kd}}=x\},$

which is the subfield of order $p^{kd}$ named by the first statement; and $kd$ runs exactly over the divisors $e$ of $m$ that are multiples of $k$ as $d$ runs over the divisors of $n$. So the intermediate fields of $F\mathrm{/}{\mathbb{F}}_q$ are precisely those subfields of $F$ whose order is $p^e$ with $k\mid e$, which is the expected answer: a subfield of $F$ contains the unique subfield of order $p^k$ exactly when $k$ divides $e$, by the divisibility clause of The intermediate fields of ${\mathbb{F}}_{q^n}\mathrm{/}{\mathbb{F}}_q$ are the ${\mathbb{F}}_{q^d}$, one for each positive divisor $d$ of $n$ applied over the prime field.

Neither statement is the other. The published one is elementary: it counts roots of $t^{p^e}-t$ and needs no Galois theory. The one proved here reads the lattice off the subgroup lattice of a cyclic Galois group, and it is that reading which the rest of this page uses, because the same correspondence also supplies the degrees and the automorphism groups of the intermediate fields. Recording their agreement here is what keeps the two vocabularies from drifting apart in later proofs.