Theorem

If $\mathbb{F}$ is a finite field, then it must have a characteristic $p$ for some prime $p$. The collection of elements $$\{j \cdot 1 : 1 \le j \le p\}$$is isomorphic to $\mathbb{F}_{p}$ and is a subfield of $\mathbb{F}$. We may therefore view $\mathbb{F}$ as a vector space over $\mathbb{F}_{p}$ which must necessarily have finite dimension. In other words, any finite field $\mathbb{F}$ must have cardinality $p^{n}$ for some prime $p$ and some natural number $n$.