Showing existence of a field extension of degree $n$ for a finite field $F$ EDIT: Just mentioning that this is a homework question.
This is my first time posting a question on math.stackexchange, so I hope you find it in your hearts to forgive any stylistic or rule transgressions I make. I have searched through quite a few of the similar threads that popped up but nothing answered my question.
The problem is as the title suggests; given a finite field $F$ and some $n > 0$, show that it has a finite field extension of degree $n$.
My attempt at a solution is as follows:
Let $|F| = p^{m}$.
Consider the splitting field of $x^{p^{mn}}-x$ over the integers modulo $p$ for some prime $p$; call it $G$. This is a finite field of order $p^{mn}$. Then it contains a subfield of size $p^{m}$, say $G'$. This is isomorphic to $F$. However, I am pretty sure $G$ does not constitute an extension of $F$.
I have tried constructing a field extension of $F$ isomorphic to $G$ by considering the image of a map $\varphi: G \rightarrow Im(\varphi)$ such that $\varphi$ restricted to G' is the isomorphism from $G$ to $F$, but I hit a wall there in showing that it was an isomorphism (briefly, I consider a n-basis for $G$ and tried defining it accordingly but wasn't able to complete it because I could not prove bijectivity).
If it is not too much trouble, I would simply prefer a tiny hint that pushes me in a promising direction. 
Thanks!
 A: Why do you think $\,G\,$ doesn't constitute an extension of your original $\,F\,$? Because it actually is, or perhaps more formally: $\,G\,$ is a vector space of dimension $\,mn\,$ over the prime field of characteristic $\,p\,\,,\,\Bbb F_p\,$ , and $\,F\,$ is a linear space of dimension $\,m\,$ over the same prime subfield. 
Let $K\leq G\,$ be a subfield of dimension $\,m\,$ over $\,\Bbb F_p\,$ . But it's not hard to show both $\,H\,\,,\,F$ are splitting  fields over $\,\Bbb F_p\,$ of the same polynomial, namely $\,x^{p^m}-x\in\Bbb F_p[x]\,$ and, thus, they're isomorphic as fields, not only as vector spaces of the same dimension over the same field.
We, in fact, have just passed above over one of the proofs (or the proof) that there's only one field of a given finite cardinality u[p to isomorphism.
Well, there you have your extension of $\,F\,$...which, in fact, you did find.
Added: I couldn't see how to give "just a hint" as the OP already solved the problem. What was lacking is to get her/him convinced that she/he actually did solve the problem and, hopefully, the above will help.
