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
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
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