The Lecture Notes (and lecture as well) to Math E-222 states the following (I've reworded a little bit):
Theorem: Let $F$ be a finite field. Then $|F|=p^n$ for some prime $p$.
Proof: Consider the canonical map $f:\mathbb{Z}\to F$. Since $\mathbb{Z}$ is infinite and $F$ finite, $\text{ker}f\not=(0)$. So $\text{ker}f=(p)$ and $\mathbb{Z}/p\mathbb{Z}\cong im(f)$ by the first isomorphism theorem. This gives F the structure of a vectorspace over the field $\mathbb{Z}/p\mathbb{Z}$ which has finite dimension, say $n$. Therefore $|F|=p^n$.
Why does F have the structure of a vectorspace? At first I thought to use Lagrange's theorem, but I think this would only show that $|F|=pn$.
I think what this is ultimately trying to show is that $F\cong F[x]/(p, x^n)$. In the case where $\text{im}(f) = F$, then it is clearly true (with restriction $x=0$), but I'm not sure what to say about those elements which are not in the image of f.
I thought that maybe we could consider the cosets of $\text{im}(f)$, but this has lead me nowhere.
Am I on the right track? Not even close? Missing something obvious?