Division algebra
Encyclopedia
In the field of mathematics
called abstract algebra
, a division algebra is, roughly speaking, an algebra over a field
, in which division
is possible.
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 algebra
s, 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).
s, 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 number
s (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
.
Over an algebraically closed field
K (for example the complex number
s C), there are no finite-dimensional associative division algebras, except K itself of course.
Associative division algebras have no zero divisor
s. 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, 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 algebra
s (see also quaternion
s).
For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology
. See for example normed division algebra
s and Banach algebra
s.
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
The following is known about the dimension of a finite-dimensional division algebra A over a field K:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
called abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a division algebra is, roughly speaking, an algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
, in which division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
is possible.
Definitions
Formally, we start with an algebraAlgebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
D over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, 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 algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
s, the definition can be simplified as follows: an associative algebra over a field is a division algebra if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it has a multiplicative identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
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 numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, which are finite-dimensional as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the reals). The Frobenius theorem
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers...
states that up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
there are three such algebras: the reals themselves (dimension 1), the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s (dimension 2), and the quaternions (dimension 4).
Wedderburn's little theorem
Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
states that if D is a finite division algebra, then D is a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
.
Over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
K (for example the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C), there are no finite-dimensional associative division algebras, except K itself of course.
Associative division algebras have no zero divisor
Zero divisor
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. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s. A finite-dimensional unital associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
(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
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F and S is a simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
over A, then the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of S is a division algebra over F; every associative division algebra over F arises in this fashion.
The center
Center (algebra)
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. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
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
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, 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
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
of the field F.
One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebra
Quaternion algebra
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...
s (see also quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s).
For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. See for example normed division algebra
Normed division algebra
In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:\|xy\| = \|x\| \|y\| for all x and y in A....
s and Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
s.
Not necessarily associative division algebras
If the division algebra is not assumed to be associative, usually some weaker condition (such as alternativityAlternativity
In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if y = x for all x and y in G and right alternative if y = x for all x and y in G...
or power associativity
Power associativity
In abstract algebra, power associativity is a property of a binary operation which is a weak form of associativity.An algebra is said to be power-associative if the subalgebra generated by any element is associative....
) is imposed instead. See algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
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
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
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
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
theorem, and was proved in 1940. The proof uses methods from topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Although a later proof was found using algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, no direct algebraic proof is known. The fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
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
Michel Kervaire
Michel André Kervaire was a French mathematician who made significant contributions to topology and algebra. He was the first to show the existence of topological n-manifolds with no differentiable structure , and computed the number of exotic spheres in dimensions greater than four...
and John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
in 1958, again using techniques of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, in particular K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. Adolf Hurwitz
Adolf Hurwitz
Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...
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 alternativeAlternative algebraIn abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have*x = y*x = y...
.
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.