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!