---
tag: 0EBE031B8E
kind: statement
role: theorem
name: A finite integral domain is a field
msc: [13G05]
uses: [09S3PFBEPM, 0QA8CHH8MG]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_domain
    locator: Properties
---

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] that has a finite number of elements.
Then $R$ is a [[field|0QA8CHH8MG]].

## Remarks

The proof: for $a \neq 0$, the map $x \mapsto a x$ from $R$ to $R$ is injective by [[cancellation|0GJM24EXP9]]. 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|0NV6E43P4A]] 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 $a$.