Formally, we start with an algebra D over a field, and assume that D does not just consist of its zero element. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb.
For associative algebras, the definition can be simplified as follows: an associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1≠0 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).
Associative division algebras
The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). The Frobenius theorem states that 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).
Whenever A is an associative unital 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 associative division algebra over F arises in this fashion.
The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F.
Not necessarily associative division algebras
If the division algebra is not assumed to be associative, usually some weaker condition (such as alternativity or power associativity) is imposed instead. See algebra over a field for a list of such conditions.
Over the reals there are (up to isomorphism) only two unitary commutative finite-dimensional division algebras: the reals themselves, and the complex numbers. These are of course both associative. For a 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 division 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. The fundamental theorem of algebra is a corollary of Hopf's theorem.
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, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8. This was independently proved by Michel Kervaire and John Milnor in 1958, again using techniques of algebraic topology, in particular K-theory. Adolf Hurwitz had shown in 1898 that the identity held only for dimensions 1, 2, 4 and 8. (See Hurwitz's theorem.)
While there are infinitely many non-isomorphic real division algebras of dimensions 2, 4 and 8, one can say the following: any real finite-dimensional division algebra over the reals must be
- isomorphic to R or C if unitary and commutative (equivalently: associative and commutative)
- isomorphic to the quaternions if noncommutative but associative
- isomorphic to the octonions if non-associative but alternative.
The following is known about the dimension of a finite-dimensional division algebra A over a field K:
- dim A = 1 if K is algebraically closed,
- dim A = 1, 2, 4 or 8 if K is real closed, and
- If K is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras over K.
- Cohn, Paul Moritz (2003). Basic algebra: groups, rings, and fields. Springer. ISBN 978-1-85233-587-8.
- Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics 131 (2 ed.). Springer. ISBN 0-387-95183-0.