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Algebra over a field with only invertible elements and zero
In abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. The multiplication
Division_algebra
Non-associative algebras with positive-definite quadratic form
is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra. If A is a Euclidean Hurwitz algebra and a is in A, define the involution
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Arithmetic operation
left-distributive and right-distributive, and thus distributive. Division is often shown in algebra and science by placing the dividend over the divisor with
Division_(mathematics)
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Generalization of quaternions to other fields
quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending
Quaternion_algebra
Ring that is also a vector space or a module
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center
Associative_algebra
Theorem in abstract algebra
abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
Type of algebras, possibly non associative
N(x)=xx^{*}} is called the norm of the algebra. A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of
Composition_algebra
Particular kind of algebraic structure
mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex
Banach_algebra
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Algebraic structure also called skew field
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial
Division_ring
Four-dimensional number system
division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra.
Quaternion
Finite dimensional algebra over a field whose central elements are that field
finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Given two central simple algebras A ~ M(n,S) and B ~
Central_simple_algebra
numbers C. The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic
Octonion_algebra
Set with operations obeying given axioms
universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure
Algebraic_structure
Branch of mathematics
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems
Algebra
Algebra where x(xy)=(xx)y and (yx)x=y(xx)
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x
Alternative_algebra
Completion of the usual space with "points at infinity"
definition, which is more often encountered in modern textbooks. Using linear algebra, a projective space of dimension n is defined as the set of the vector
Projective_space
Algebraic manipulation of "true" and "false"
mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables
Boolean_algebra
Abelian group related to division algebras
to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form
Brauer_group
In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras
Okubo_algebra
associative division algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the
Exceptional_object
Algebra over a field where binary multiplication is not necessarily associative
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative
Non-associative_algebra
Optimal stable 2-systolic inequality
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality s t s y s 2 n ≤ n ! v o l 2 n ( C P n ) {\displaystyle \mathrm
Gromov's inequality for complex projective space
Gromov's_inequality_for_complex_projective_space
Method for producing composition algebras
division algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative division
Cayley–Dickson_construction
Element of a unital algebra over the field of real numbers
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
Hypercomplex_number
Algebraic structure with addition, multiplication, and division
multiplication, and division are defined and behave as the corresponding operations on rational numbers do. Fields are fundamental algebraic structures that
Field_(mathematics)
Hypercomplex number system
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented
Octonion
Concept in ring theory
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R {\displaystyle R} -algebras where R {\displaystyle R} need not
Azumaya_algebra
Classification of semi-simple rings and algebras
for algebras over a field k. If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra
Wedderburn–Artin_theorem
Canadian mathematician and physicist
Volkswagen Foundation. Her main interests are division algebras, Clifford algebras, and Jordan algebras, and their relation to particle physics. Her work
Cohl_Furey
Homomorphisms between simple modules over the same ring are isomorphisms or zero
elementary but useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible
Schur's_lemma
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
Classification in abstract algebra
are (non-canonically) isomorphic. The dimensions of the matrix algebra, and what division ring (R, C, H) can be determined by the dimension of the vector
Classification of Clifford algebras
Classification_of_Clifford_algebras
Setting of relativistic physics in geometric algebra
spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) of physics. Spacetime algebra provides
Spacetime_algebra
In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple
Cyclic_algebra
Result pertaining to division rings
abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It
Cartan–Brauer–Hua_theorem
Relation between Lie algebras depicted as a square
idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table
Freudenthal_magic_square
German mathematician (1882–1935)
German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental
Emmy_Noether
Natural number
cube-octahedron compound. The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers. They are a double cover
8
Academic journal
Communications in Algebra is a monthly peer-reviewed scientific journal covering algebra, including commutative algebra, ring theory, module theory, non-associative
Communications_in_Algebra
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
Overview of and topical guide to algebraic structures
types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures
Outline of algebraic structures
Outline_of_algebraic_structures
b, c such that if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions of isotopy are equivalent
Isotopy_of_an_algebra
Distance from zero to a number
value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example, a
Absolute_value
Concept in mathematics
determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore (1922). Because quaternion multiplication does not
Moore determinant of a Hermitian matrix
Moore_determinant_of_a_Hermitian_matrix
American mathematician (1905–1972)
in Algebra for his work on Riemann matrices. He is best known for his work on the Albert–Brauer–Hasse–Noether theorem on finite-dimensional division algebras
A._A._Albert
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is
Geometric_algebra
In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related
Biquaternion_algebra
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
GLdr(F), where D is a division algebra of degree d2 over the local or global field F. Suppose that G is an inner twist of the algebraic group GL2, in other
Jacquet–Langlands correspondence
Jacquet–Langlands_correspondence
Number with a real and an imaginary part
solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex
Complex_number
Result in algebra
theorem by the following argument. Let D {\displaystyle D} be a finite division algebra with center k {\displaystyle k} . Let [ D : k ] = n 2 {\displaystyle
Wedderburn's_little_theorem
Topics referred to by the same term
Frobenius theorem (real division algebras) in abstract algebra characterizing the finite-dimensional real division algebras Frobenius reciprocity theorem
Frobenius_theorem
Mathematical symbol
one or more dots, was first used as a symbol for division in 1659, in the algebra book Teutsche Algebra by Johann Rahn, although previous writers had used
Division_sign
Geometric space with eight dimensions
dimensions is 240. The octonions are a normed division algebra over the real numbers, the largest such algebra. Mathematically they can be specified by 8-tuplets
Eight-dimensional_space
Scottish mathematician
proved that a finite division algebra is a field (Wedderburn's little theorem), and part of the Artin–Wedderburn theorem on simple algebras. He also worked
Joseph_Wedderburn
Branch of mathematics that studies algebraic structures
algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures
List of abstract algebra topics
List_of_abstract_algebra_topics
Scientific area at the interface between computer science and mathematics
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the
Computer_algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until
History_of_algebra
Quaternions with complex number coefficients
The algebra of biquaternions can be considered as a tensor product C ⊗R H, where C is the field of complex numbers and H is the division algebra of (real)
Biquaternion
Associative Artinian algebra with a trivial Jacobson radical
semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson
Semisimple_algebra
Magma obeying the Latin square property
mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible. Quasigroups
Quasigroup
Representation of a group or algebra in terms of an algebra with quaternionic structure
of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation
Quaternionic_representation
Algebraic structure
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more
Polynomial_ring
American mathematician (1902–1997)
earned her doctorate in 1931 with a dissertation on abstract algebra titled "Division algebras associated with an equation whose group has four generators
Mina_Rees
Basic concepts of algebra
{b^{2}-4ac}}}{2a}}}}}} Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted
Elementary_algebra
Algebraic structure with a binary operation
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with
Magma_(algebra)
Algebraic structure
noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative
Noncommutative_ring
mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Theorem in number theory
In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits
Albert–Brauer–Hasse–Noether theorem
Albert–Brauer–Hasse–Noether_theorem
Eight-dimensional algebra over the real numbers
commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra. In mechanics, the dual quaternions
Dual_quaternion
Mathematical concept
normed division algebras are only possible in 1, 2, 4 and 8 dimensions. The cross product is formed from the product of the normed division algebra by restricting
Seven-dimensional cross product
Seven-dimensional_cross_product
isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers C {\displaystyle \mathbb {C} } . The
Gelfand–Mazur_theorem
Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
Israeli mathematician (1921–1994)
including the theory of rings with polynomial identities (PI-rings), division algebras, the general theory of radicals, and the Amitsur complex in descent
Shimshon_Amitsur
American mathematician
mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is
Leonard_Eugene_Dickson
Algebraic structure in linear algebra
geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and
Vector_space
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
Open convex self-dual cones
type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other
Symmetric_cone
= 1. Banach algebra Composition algebra Hurwitz's theorem (composition algebras) Division algebra Gelfand–Mazur theorem "Normed Algebra". Encyclopaedia
Normed_algebra
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smooth
Vector_fields_on_spheres
Method in wireless communication systems
form of estimation. These codes have been studied more widely, and division algebras over number fields have now become the standard tool for constructing
Space–time_code
Algebra where division is always defined
A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The
Wheel_theory
Mathematical operation
multiplication, division, raising to a whole number power, and taking roots (fractional power). The operations of elementary algebra may be performed
Algebraic_operation
Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both
Commutative_algebra
Type of ring in non-commutative algebra
finite-dimensional simple algebra over a field k {\displaystyle k} , it is isomorphic to a matrix algebra over some division algebra over k {\displaystyle
Simple_ring
In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field
Separable_algebra
Vector on which a quadratic form is zero
inverse for x, and since x ≠ 0, A is not a division algebra. In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers,
Null_vector
ISBN 978-0-387-72487-4. Zbl 1130.12001. Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874
Factor_system
Non-tensorial representation of the spin group
spin group or of the associated Clifford algebra. After choosing a matrix realization of the Clifford algebra, spinors may be represented concretely as
Spinor
Tensor product of algebras over a field; itself another algebra
the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field
Tensor_product_of_algebras
Algorithm for Euclidean division of polynomials
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long
Synthetic_division
1966 mathematics textbook by Serge Lang
focus and is aimed at upper-division undergraduates who have been exposed to linear algebra in a prior course. Linear Algebra is designed for a one-semester
Linear_Algebra_(book)
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
Free object in the category of associative algebras
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since
Free_algebra
Branch of algebra
language, modules; special classes of rings (group rings, division rings, universal enveloping algebras); related structures like rngs; as well as an array
Ring_theory
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Mathematical ring whose elements are matrices
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The
Matrix_ring
Mathematical operation on vectors in 3D space
related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8. In general dimension,
Cross_product
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