The order of a finite field is a prime power

Statement

Let $F$ be a finite field. Then the characteristic of $F$ is a prime number $p$, and $|F| = p^n$ with $n = [F : \mathbb{F}_p]$.

Remarks

The proof: a finite field has nonzero characteristic, and the characteristic of a domain is $0$ or prime. The prime subfield is $\mathbb{F}_p$, and $F$ is a vector space of dimension $n$ over it, so it has $p^n$ elements. So there is no field with $6$ or $10$ elements. The converse, that a field of order $p^n$ exists for each $n \ge 1$, is the existence theorem.

Explanations

L1 human

A finite field is a vector space over the field with $p$ elements. A vector space of dimension $n$ over it has $p^n$ elements. So the size of a finite field is always a power of a prime.