How many subfields of order $p^m$ in $\mathbb{F}_{p^n}$

Question

We know there is exists a subfield of order $p^m$ in $\mathbb{F}_{p^n}$ if $m$ divides $n$. My questions is if $m$ does not divide $n$, is there still such a subfield? And how many subfields are there for each $m|n$?

Answer 1

$\operatorname{Gal}(\Bbb F_{p^n}/\Bbb F_p) = \Bbb Z/n\Bbb Z$

Using the Fundamental theorem of Galois theory, one sees that there is only one such subfield if $m \mid n$, and zero otherwise.

---

A more elementary proof that the existence of a subfield of order $p^m$ implies $m \mid n$ is to note that if $E$ is such a subfield, then $m = [E : \Bbb F_p] \mid [\Bbb F_{p^n} : E] [E : \Bbb F_p] = [\Bbb F_{p^n} : \Bbb F_p] = n$.