Division algebra

In abstract algebra, a division algebra is a unitary associative algebra with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).

Some authors omit the associativity requirement and define a division algebra to be an algebra D over a field such that for any element a in D and any non-zero element b in D there exists precisely one element x with a = bx and precisely one element y in D such that a = yb.

Table of contents

1 Associative division algebras 2 Not neccessarily associative division algebras 3 See also

Associative division algebras

In this section we assume the division algebras are associative.

The best-known examples of division algebras are the finite-dimensional real division algebras (that is, division algebras over R (the field of real numbers), which are finite-dimensional as a vector space over the reals). Up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4). This was proved by Frobenius in 1877.

There is no finite-dimensional division algebra over the complex numbers (except for the complex numbers themselves).

Division algebras have no zero divisors. A finite-dimensional algebra is a division algebra if and only if it has no zero divisors.

Every field extension forms a division algebra over the ground field.

Whenever A is an associative algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every division algebra over F arises in this fashion.

Not neccessarily associative division algebras

If the division algebra is not assumed to be associative, usually some weaker associativity condition is imposed instead. See algebra over a field.

Over the reals there are (up to isomorphism) only two commutative finite-dimensional division algebras with unit: the reals themselves, and the complex numbers. These are of course both associative. For an non-associative example, consider the complex numbers with multiplication defined by taking the complex conjugate of the usual multiplication. This is a commutative, non-associative algebra of dimension 2 over the reals, and has no unit element. There are infinitely many other non-isomorphic commutative, non-associative, finite-dimensional real divisional algebras, but they all have dimension 2.

In fact, every finite-dimensional real commutative division algebra is either 1 or 2 dimensional. This is known as Hopf's theorem, and was proved in 1940. The proof uses methods from topology. Although a later proof was found using algebraic geometry, no direct algebraic proof is known.

Dropping the requirement of commutativity, Hopf generalized his result: Any finite-dimensional real division algebra must have dimension a power of 2.

Later work showed that in fact, for any finite-dimensional division algebra must be of dimension 1, 2, 4, or 8. This was independently proved by Kervaire and Milnor in 1958, again using techniques of algebraic topology, in particular K-theory. In the case of the reals, the four possible division algebras are the reals, complexes, quaternions, and octonions.

Note also that the fundamental theorem of algebra is a corollary of Hopf's theorem.

Every alternative division algebra has a unit element. Thus all alternative 2-dimensional real division algebras are isomorphic to the complex numbers.

For infinite-dimensional division algebras, the most important cases are those where the space has some reasonable topology. See for example normed division algebras.

See also

See also: normed division algebra, division, division ring