Finite fields

Characteristic. The characteristic of a finite field $F$ is $p$ (prime): otherwise, $\mathbb{Z}/n\mathbb{Z}\subseteq F$ for composite $n$, which has zero divisors, contradicting that $F$ is a field.

Vector space. $F$ contains $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$, and is a finite-dimensional $\mathbb{F}_p$-vector space of dimension $n=[F:\mathbb{F}_p]$.

Order. $|F|=|\mathbb{F}_p|^n=p^n$.

Conclusion. Every finite field has order $p^n$. Conversely, for every prime power $p^n$, there exists a finite field of order $p^n$ (unique up to isomorphism, denoted $\mathbb{F}_{p^n}$).