Wedderburn's little theorem

Every finite division ring is commutative and therefore a field. The theorem rules out finite noncommutative division rings.

Division ring: A ring in which every nonzero element has a multiplicative inverse, without requiring multiplication to be commutative. The theorem's starting structure allows noncommutative multiplication.
Class equation: An identity expressing a finite group's order as the sum of the sizes of its conjugacy classes. Applied to the nonzero elements, it organizes the count by centralizers.
Joseph Wedderburn: A Scottish mathematician whose work helped establish the structure theory of associative algebras. His 1905 proof gave the theorem its familiar name.
Finite projective plane: A projective plane with finitely many points and lines, satisfying the usual incidence axioms. Finite coordinate constructions cannot use noncommutative division rings.
Center of a ring: The set of elements in a ring that commute with every element of the ring. The proof studies the division ring through its center, which is a finite field.
Centralizer: The subgroup of elements commuting with a specified element or set of elements. Noncentral elements are counted using centralizers that are smaller division rings.
Leonard Eugene Dickson: An American mathematician who made foundational contributions to finite fields, groups, and algebras. He independently proved the result in 1905.
Projective plane: A geometric incidence structure in which every pair of points determines a line and every pair of lines meets. The theorem restricts algebraic coordinate systems available for finite examples.
Conjugacy class: The set of elements obtained from one group element by conjugation by every member of the group. Counting conjugacy classes in the multiplicative group is central to a standard proof.
Polynomial over a field: A polynomial whose coefficients lie in a field, with roots considered in that field or an extension. The proof bounds how many elements can satisfy equations over the center.

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