Division algebra - Citizendia

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

1 Definitions
2 Associative division algebras
3 Not necessarily associative division algebras
4 References
5 See also

Definitions

Formally, we start with an algebra D over a field, and assume that D does not just consist of its zero element. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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 iff it has a multiplicative identity element 1≠0 and every non-zero element a has a multiplicative inverse (i. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive ↔ In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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). In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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). In Mathematics, more specifically in Abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877 characterizes the finite dimensional In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements (T. Y. Lam, A First Course in Noncommutative Rings. )

Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself of course. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Associative division algebras have no zero divisors. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i

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. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, a (left or right module S over a ring R is called simple or irreducible if it is not the Zero In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object

The center of an associative division algebra D over the field K is a field containing K. The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements 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. In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other Given a field F, the (isomorphism classes) of 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. In Mathematics, the Brauer group arose out of an attempt to classify Division algebras over a given field K.

One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions). In Mathematics, a quaternion algebra over a field F, is a particular kind of Central simple algebra, A, over F, namely such an algebra Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of See for example normed division algebras and Banach algebras. In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

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. In Abstract algebra, a magma G is said to be left alternative if ( xx) y = x ( xy) for all x and y In Abstract algebra, power associativity is a weak form of Associativity. See algebra over a field for a list of such conditions. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with

Over the reals there are (up to isomorphism) only two unitary commutative finite-dimensional division algebras: the reals themselves, and the complex numbers. In Mathematics, commutativity is the ability to change the order of something without changing the end result 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:

$a*b=\overline{ab}.$

This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. This page presents and discusses an example of a non-associative Division algebra over the Real numbers The multiplication is defined by taking the Complex conjugate 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. Heinz Hopf ( November 19, 1894 – June 3, 1971) was a German Mathematician born in Gräbschen, Germany (now The proof uses methods from topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Although a later proof was found using algebraic geometry, no direct algebraic proof is known. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with The fundamental theorem of algebra is a corollary of Hopf's theorem. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at

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. Michel André Kervaire ( Częstochowa, Poland, 26 April, 1927 &ndash Geneva, Switzerland, 19 November, John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, K-theory is a tool used in several disciplines Adolf Hurwitz had shown in 1898 that the identity $q\overline{q} = \textrm{sum\ of\ squares}$ held only for dimensions 1, 2, 4 and 8. Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre (See Hurwitz's theorem. In Mathematics, Hurwitz's theorem is any of at least five different results named after Adolf Hurwitz. ) ^[1]

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 non-commutative but associative
isomorphic to the octonions if non-associative but alternative. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only 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. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, a real closed field is a field F in which any of the following equivalent conditions are true There is a Total order

References

^ Roger Penrose (2005). Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus The Road To Reality. Vintage. ISBN 0-09-944068-7.

Contents

Definitions

Associative division algebras

Not necessarily associative division algebras

References

See also