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 Classification
Statement Form Connections
Has Statement Form
In Branch
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.
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