TITLE: Proving that $GF(p^n)$ contains a unique subfield isomorphic to $GF(p^m)$ if and only if $m$ is a divisor of $n$.
QUESTION [5 upvotes]: I am looking to prove that $GF(p^n)$ contains a unique subfield isomorphic to $GF(p^m)$ if and only if $m$ is a divisor of $n$.
In the Wikipedia article:
https://en.wikipedia.org/wiki/Finite_field#Existence_and_uniqueness
They say that this statement has been proven by E. H. Moore in 1893.
How to go about proving the theorem? I know first I need to generate a subset, prove that it's a subfield, and then assume they are two and show it's unique.
But how to generate it in the first place? Consider the set of polynomials of the form $x^{p^m}-x=0$? Is there a way to show that it's a subfield without checking all the axioms?

REPLY [2 votes]: Use the Galois correspondence.   $GF(p^n)$ is a Galois extension of $GF(p)$, which has cyclic Galois group.  
(Intermediate) extension fields correspond to subgroups (of the Galois group).  We have a subgroup of order $m\iff m\mid n\iff$ we have an extension field of degree $m$.