In the field of mathematics called abstract algebra, a **division algebra** is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.

## . . . Division algebra . . .

Formally, we start with a non-zeroalgebra*D* over a field. 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: a non-zero associative algebra over a field is a **division algebra**if and only if it has a multiplicative identity element 1 and every non-zero element *a* has a multiplicative inverse (i.e. an element *x* with *ax* = *xa* = 1).

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 toisomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).

Wedderburn’s little theorem states that if *D* is a finite division algebra, then *D* is a finite field.[1]

Over an algebraically closed field*K* (for example the complex numbers**C**), there are no finite-dimensional associative division algebras, except *K* itself.[2]

Associative division algebras have no zero divisors. A *finite-dimensional*unitalassociative algebra (over any field) is a division algebra *if and only if* it has no zero divisors.

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*.

One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).

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

## . . . Division algebra . . .

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