Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$.
Theorem: Let $E$ be a degree $n$ finite extension of a field $F$. If $F$ has $q$ elements, then $E$ has $q^n$ elements.
Definition: Let $E$ be a field. Suppose there exists $n\geq 1$ such that $n\cdot x = 0$ for all $x\in E$. Then the smallest such choice of $n$ is the characteristic of $E$. If no such $n$ exists, then $E$ is of characteristic $0$.
Corollary: Let $E$ be a finite field with characteristic $p$. Then $E$ contains exactly $p^n$ elements for some choice of $n\geq 1$.
Proof: (Taken from Fraleigh - A First Course in Algebra, 7Ed) Every finite field $E$ is a finite field extension of a prime field isomorphic to the field $\mathbb{Z}_{p}$, where $p$ is the characteristic of $E$. The result follows from the theorem using $F = \mathbb{Z}_{p}$.
This proof is probably very simple, but I'm having problems with showing that the degree of $E$ over $\mathbb{Z}_{p}$ is finite. It seems intuitively obvious since $E$ itself is finite, but I cannot see how to conclude from this that $E$ is a finite-dimensional vector space over $\mathbb{Z}_{p}$. Obviously it is not an infinite-dimensional vector space over $F$ since it is not infinite.
This may seem too simple to answer, but what am I missing here?
Thanks for any assistance!
