Division algebra

A division algebra is an associative algebra with identity whose every nonzero element is invertible; when commutativity is not assumed, it is also called a division ring.² Its center consists of elements that commute with everything, and this center is a field. Consequently, any division algebra can be regarded as an algebra over its center, but it need not be finite-dimensional over that field.

Fields are precisely the commutative division algebras. Noncommutative examples retain many familiar cancellation properties: if ab = ac with a nonzero, then b = c. The quaternions form the standard finite-dimensional example, with relations i² = j² = k² = ijk = −1.¹

Finite-dimensional division algebras are often organized by their center and by their dimension over it. A central division algebra over a field F has center exactly F; its dimension over F is necessarily a square, commonly written n², where n is its degree.

The classification is simple over some fields and highly intricate over others. Frobenius' theorem states that the only finite-dimensional associative division algebras over the real numbers are the real numbers, complex numbers, and Hamilton's quaternions. By contrast, central division algebras over number fields, local fields, and function fields produce rich arithmetic families. Their equivalence classes are recorded by the Brauer group, whose operation is induced by tensor product and whose identity class consists of matrix algebras over the base field.

Every finite division ring is commutative, so finite division rings are exactly finite fields; this is Wedderburn's little theorem.² The result is striking because noncommutativity is possible in infinite division rings and in finite-dimensional algebras over infinite fields, yet finiteness alone forces commutativity.

Important constructions include opposite algebras, tensor products, and quotients of suitable noncommutative polynomial rings. A central simple algebra may be a full matrix algebra over a field without itself being a division algebra; when it is replaced by the unique division algebra in its Brauer-equivalence class, its noncommutative structure becomes more visible. Maximal subfields inside these algebras provide a bridge between field theory, representation theory, and arithmetic.

The word “division” does not imply that every algebra called a division algebra is associative: octonions have inverses for nonzero elements but are nonassociative, so they belong to the broader theory of alternative algebras rather than to the usual division-ring definition.¹ This distinction separates the Frobenius classification from the Hurwitz classification of finite-dimensional real normed division algebras.

Division algebras also appear beyond pure classification. Quaternions encode rotations and are used in computer graphics, robotics, and spacecraft attitude calculations, while central simple algebras and Brauer groups occur in algebraic number theory and the study of algebraic varieties. A further edge case is that a division algebra can be infinite-dimensional over its center, so finite-dimensional techniques do not describe the whole subject.

division ring

An associative ring with identity in which every nonzero element has a multiplicative inverse.

center

The subfield of elements in a division algebra that commute with every element.

central simple algebra

A finite-dimensional associative algebra that is simple and has a prescribed field as its center.

Brauer group

A group of equivalence classes of central simple algebras over a field, with tensor product as the operation.

quaternion

An element of Hamilton's four-dimensional noncommutative division algebra over the real numbers.