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.

Formally, we start with a non-zero algebra *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 to isomorphism 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* unital associative 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.

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.^{ [3] } (See Hurwitz's theorem.) The challenge of constructing a division algebra of three dimensions was tackled by several early mathematicians. Kenneth O. May surveyed these attempts in 1966.^{ [4] }

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*.

- ↑ Lam (2001), p. 203
- ↑ Cohn (2003), Proposition 5.4.5, p. 150
- ↑ Roger Penrose (2005).
*The Road To Reality*. Vintage. ISBN 0-09-944068-7., p.202 - ↑ Kenneth O. May (1966) "The Impossiblility of a Division Algebra of Vectors in Three Dimensional Space", American Mathematical Monthly 73(3): 289–91 doi : 10.2307/2315349

In mathematics, an **associative algebra** is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give *A* the structure of a ring; the addition and scalar multiplication operations together give *A* the structure of a vector space over *K*. In this article we will also use the term *K*-algebra to mean an associative algebra over the field *K*. A standard first example of a *K*-algebra is a ring of square matrices over a field *K*, with the usual matrix multiplication.

In mathematics, especially functional analysis, a **Banach algebra**, named after Stefan Banach, is an associative algebra *A* over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

In abstract algebra, a **division ring**, also called a **skew field**, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with *a*·*x* = *x*·*a* = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where *noncommutative ring* refers to rings which are not *necessarily* commutative.

In mathematics, a **field** is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, a **ring** is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

In mathematics, **hypercomplex number** is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

In mathematics, the **quaternions** are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.

In ring theory, a branch of abstract algebra, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In mathematics, the **Cayley–Dickson construction**, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as **Cayley–Dickson algebras**, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

In mathematics, an **algebra over a field** is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

In algebra, **ring theory** is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In mathematics, the **Brauer group** of a field *K* is an abelian group whose elements are Morita equivalence classes of central simple algebras over *K*, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

In ring theory and related areas of mathematics a **central simple algebra** (**CSA**) over a field *K* is a finite-dimensional associative *K*-algebra*A*, which is simple, and for which the center is exactly *K*. As an example, note that any simple algebra is a central simple algebra over its center.

In abstract algebra, a branch of mathematics, a **simple ring** is a non-zero ring that has no two-sided ideal besides the zero ideal and itself.

In abstract algebra, a **matrix ring** is any collection of matrices over some ring *R* that form a ring under matrix addition and matrix multiplication. The set of *n* × *n* matrices with entries from *R* is a matrix ring denoted M_{n}(*R*), as well as some subsets of infinite matrices which form **infinite matrix rings**. Any subring of a matrix ring is a matrix ring.

In mathematics, a **composition algebra**A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies

A **non-associative algebra** is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure *A* is a non-associative algebra over a field *K* if it is a vector space over *K* and is equipped with a *K*-bilinear binary multiplication operation *A* × *A* → *A* which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (*ab*)(*cd*), *d* and *a*(*b* ) may all yield different answers.

In mathematics, the **field with one element** is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted **F**_{1}, or, in a French–English pun, **F**_{un}. The name "field with one element" and the notation **F**_{1} are only suggestive, as there is no field with one element in classical abstract algebra. Instead, **F**_{1} refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of **F**_{1} have been proposed, but it is not clear which, if any, of them give **F**_{1} all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.

In mathematics, more specifically abstract algebra and ring theory, a **noncommutative ring** is a ring whose multiplication is not commutative; that is, there exists *a* and *b* in *R* with *a***·***b* ≠ *b***·***a*. Many authors use the term *noncommutative ring* to refer to rings which are not necessarily commutative, and hence include commutative rings in their definition. **Noncommutative algebra** is the study of results applying to rings that are not required to be commutative. Many important results in the field of noncommutative algebra area apply to commutative rings as special cases.

- Cohn, Paul Moritz (2003).
*Basic algebra: groups, rings, and fields*. London: Springer-Verlag. doi:10.1007/978-0-85729-428-9. ISBN 978-1-85233-587-8. MR 1935285. - Lam, Tsit-Yuen (2001).
*A first course in noncommutative rings*. Graduate Texts in Mathematics.**131**(2 ed.). Springer. ISBN 0-387-95183-0.

- "Division algebra",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.