---
tag: 0RRNVX1VKA
kind: statement
role: theorem
name: Wedderburn's little theorem
msc: ["12E15"]
uses: [0WZXK05ME7, 0QA8CHH8MG]
sources:
  - url: https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem
    locator: History
same_as:
  wikidata: Q943246
  wikipedia: Wedderburn's little theorem
---

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]] with a finite number of elements.
Then the multiplication of $D$ is commutative, so $D$ is a [[field|0QA8CHH8MG]].

## Remarks

The hypothesis of finiteness is necessary: the [[quaternions|0GE4DVQKR8]] are an infinite division ring that is not commutative.
A consequence is that every finite domain, commutative or not, is a field.

## History

Joseph Wedderburn gave the first proof in 1905.
Leonard Eugene Dickson gave another proof soon after.
The first proof of Wedderburn had a gap, and for this reason some historians credit Dickson with the first correct proof.

## Explanations

### L1 human

In an infinite division ring, $a b$ and $b a$ can be different.
The theorem says that this cannot happen when the ring is finite.
So there are no finite analogues of the quaternions.