A finite integral domain is a field

Statement

Let $R$ be an integral domain that has a finite number of elements. Then $R$ is a field.

Remarks

The proof: for $a \neq 0$, the map $x \mapsto a x$ from $R$ to $R$ is injective by cancellation. A finite set has no injective map to itself that is not surjective, so $a x = 1$ for some $x$. A consequence is that $\mathbb{Z}/p\mathbb{Z}$ is a field for every prime $p$. The hypothesis of finiteness is necessary: $\mathbb{Z}$ is an infinite integral domain that is not a field.

Explanations

L1 human

In a finite integral domain, multiplication by a nonzero element $a$ never sends two elements to the same element. Since there are only finitely many elements, every element is a product $a x$. In particular, $1 = a x$ for some $x$, and $x$ is the inverse of $a$.