Every finite division ring is a field, meaning multiplication in a finite division ring is automatically commutative. Proved by Joseph Wedderburn, the theorem is a striking example of a finiteness condition forcing an algebraic structure into a more restrictive one.
Facts
StatementEvery finite division ring is a field; equivalently, every finite integral domain is a field, so for finite rings there is no distinction between domains, division rings and fields. 1 Proof Method Classification
Statement Form Connections
Has Statement Form
In Branch
Principal Result Of
Source Wikipedia: Joseph Wedderburn
Proved By
Source Wedderburn's Little Theorem (Wikipedia)
Sources
1. Wedderburn's Little Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field.
History clause on Wedderburn's 1905 proof
The original proof was given by Joseph Wedderburn in 1905
Proved By: Joseph Wedderburn, Lead
The original proof was given by Joseph Wedderburn in 1905, who went on to prove the theorem in two other ways.
In Group: Ring Theory and Commutative Algebra, Lead section
Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field.
View the Source 2. ProofWiki: Wedderburn's Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose $m > 1$.
Wikipedia: Joseph Wedderburn
Principal Result Of: Joseph Wedderburn, Article textQuote, Principal Result Of: Joseph Wedderburn, Article text
A significant algebraist, he proved that a finite division algebra is a field (Wedderburn's little theorem), and part of the Artin-Wedderburn theorem on simple algebras.
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