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Surjective ring homomorphism with a given codomain
In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension. Precisely, a ring extension of a ring R by an abelian
Algebra_extension
Extension of a mathematical field with polynomial roots
In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that
Algebraic_extension
Creating a "larger" Lie algebra from a smaller one, in one of several ways
algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise
Lie_algebra_extension
Algebraic structure where all polynomials have roots
algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]
Algebraically_closed_field
Type of algebraic field extension
field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle
Separable_extension
Group for which a given group is a normal subgroup
mechanics. Algebra extension Lie algebra extension Virasoro algebra HNN extension Group contraction Extension of a topological group group+extension#Definition
Group_extension
Construction of a larger algebraic field by "adding elements" to a smaller field
In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L {\displaystyle K\subseteq L} , such that the operations of K are those
Field_extension
Algebraic structure with addition, multiplication, and division
operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics
Field_(mathematics)
Type of algebraic field extension
In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits
Normal_extension
Algebraic field extension
mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures
Algebraic_closure
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
Algebraic structure with "nice" duality properties
finite-dimensional extension field of k, then every k-algebra A gives rise naturally to an F algebra, F ⊗k A, and A is a Frobenius k-algebra if and only if
Frobenius_algebra
Finite extension of the rationals
In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle
Algebraic_number_field
Algebraic structure
{\displaystyle K} is separable. Every finite extension of K {\displaystyle K} is separable. Every algebraic extension of K {\displaystyle K} is separable. Either
Perfect_field
Algebraic field extension
mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed
Galois_extension
Field extension that is not algebraic
degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence
Transcendental_extension
Theorem extending pre-measures to measures
its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending μ 0 {\displaystyle \mu _{0}} from an algebra of
Carathéodory's extension theorem
Carathéodory's_extension_theorem
Algebra describing 2D conformal symmetry
mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional
Virasoro_algebra
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Type of Kac–Moody algebras
affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given
Affine_Lie_algebra
Topics referred to by the same term
Trivial extension may refer to the following types of extensions: A trivial field extension A trivial group extension A trivial algebra extension This disambiguation
Trivial_extension
Branch of mathematics
Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester
Linear_algebra
Mathematical element
notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial
Integral_element
Topics referred to by the same term
algebra, Grassmann's theory of extension, in geometry Field extension, in Galois theory Group extension, in abstract algebra and homological algebra Homotopy
Extension
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
Alebraic concept
In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of
Purely_inseparable_extension
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
the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators. This extension is called the
Toeplitz_algebra
Field theory is the branch of algebra that studies fields
factors. Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure
Glossary_of_field_theory
Generalization of quaternions to other fields
field extension), i.e. for a suitable field extension K of F, A ⊗ F K {\displaystyle A\otimes _{F}K} is isomorphic to the 2 × 2 matrix algebra over K
Quaternion_algebra
Concept in abstract algebra
are algebraic over K {\displaystyle K} , then L / K {\displaystyle L/K} is called an algebraic extension. These notions generalize the algebraic numbers
Algebraic_element
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
Algebraic structure used in analysis
Gelfand–Fuks cohomology Hopf algebra Index of a Lie algebra Leibniz algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie bialgebra
Lie_algebra
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 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
Subset of a ring that forms a ring itself
the sum of n copies of 1 equals 0. Integral extension Group extension Algebraic extension Ore extension In general, not all subsets of a ring R are rings
Subring
Topics referred to by the same term
polynomials Algebraic element, an element of a field extension which is a root of some polynomial over the base field Algebraic extension, a field extension such
Algebraic
Relates the topology of a complete non-archimedean field to its algebraic extensions
relating the topology of a complete non-archimedean field to its algebraic extensions. Let K be a complete non-archimedean field and let K be a separable
Krasner's_lemma
separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. A homomorphism
Separable_algebra
Algebraic structure in mathematics
quadratic algebra. The Weyl algebra of a finite-dimensional symplectic vector space is a filtered quadratic algebra. Algebraic element Algebraic extension Koszul
Quadratic_algebra
Category theory constructs
certain (Kan) extensions using limits in 1960. An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived
Kan_extension
Branching out of a mathematical structure
valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local
Ramification_(mathematics)
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
Galois extension whose Galois group is abelian
In algebraic number theory, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension
Abelian_extension
Type of complex number
algebraic if and only if both a and b are algebraic. For any α {\displaystyle \alpha } , the simple extension of the rationals by α {\displaystyle \alpha
Algebraic_number
Theorem in field theory
The theorem states that given any field F {\displaystyle F} , an algebraic extension field E {\displaystyle E} of F {\displaystyle F} and an isomorphism
Isomorphism_extension_theorem
Set without nontrivial polynomial equalities
In abstract algebra, a subset S {\displaystyle S} of a field L {\displaystyle L} is algebraically independent over a subfield K {\displaystyle K} if the
Algebraic_independence
algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions.
Étale_algebra
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
Algebraic structure used in logic
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with
Heyting_algebra
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
theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
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
Type of Lie algebra of interest in physics
In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. For a Lie algebra g {\displaystyle {\mathfrak
Loop_algebra
glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary
Glossary of commutative algebra
Glossary_of_commutative_algebra
Supersymmetric generalization of the Poincaré algebra
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions
Super-Poincaré_algebra
Topological complex vector space
mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties
C*-algebra
Type of algebra
algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle R} -algebra for
Finitely_generated_algebra
Mathematical group
In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends
Galois_group
Mathematical operation on vector spaces
called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule. Given two fields that are algebraic extensions of a
Tensor_product
Polynomial equation, generally univariate
for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element
Algebraic_equation
Group of flat spacetime symmetries
{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More
Poincaré_group
Branch of mathematics
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument
Multilinear_algebra
Mathematical software
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in
Computer_algebra_system
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
Computer algebra system
algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed hierarchy. Two computer algebra systems
Axiom (computer algebra system)
Axiom_(computer_algebra_system)
especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are
Ore_extension
Universal construction in multilinear algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being
Tensor_algebra
the algebra is said to have extended supersymmetry. The supersymmetry algebra is a semidirect sum of a central extension of the super-Poincaré algebra by
Supersymmetry_algebra
Supersymmetric extension to the Virasoro algebra
Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular
Super_Virasoro_algebra
Cohomology theory for Lie algebras
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of
Lie_algebra_cohomology
Construction in homological algebra
associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module
Ext_functor
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Type of field extension
field theory, a branch of algebra, a field extension L / k {\displaystyle L/k} is said to be regular if k is algebraically closed in L (i.e., k = k ^
Regular_extension
Finite dimensional algebra over a field whose central elements are that field
areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center
Central_simple_algebra
Reasoning about equations with free variables
proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators." Algebraic formalisms
Algebraic_logic
Curve defined as zeros of polynomials
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in
Algebraic_curve
union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym
Absolute_irreducibility
mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
Computational method
coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the
Factorization_of_polynomials
Algebra of meromorphic vector fields on the Riemann sphere
{\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}.} This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal
Witt_algebra
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
In mathematics, invertible homomorphism
unique. The term isomorphism is mainly used for algebraic structures and categories. In the case of algebraic structures, mappings are called homomorphisms
Isomorphism
Topics referred to by the same term
energetic material Inert prime, a type of behaviour of a prime under an algebraic extension Inert waste, waste which is neither chemically nor biologically reactive
Inert
Complex number that solves a monic polynomial with integer coefficients
following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of Q {\displaystyle \mathbb {Q} } , the field
Algebraic_integer
Study of Galois symmetry groups of differential fields
is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois
Differential_Galois_theory
Type of residuated Boolean algebra with extra structure
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation
Relation_algebra
Branch of algebra that studies commutative rings
rings (the main class of commutative rings occurring in algebraic number theory), integral extensions, and valuation rings. Polynomial rings in several indeterminates
Commutative_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 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)
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
Boolean algebra with all operators and laws forming a complete logical system
open algebra can be used to form Boolean-valued models which are then equivalent to generic extensions by the given forcing poset. The algebra of all
Complete_Boolean_algebra
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
Algebra used in 2D conformal field theories and string theory
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string
Vertex_operator_algebra
Dimension of the extension field viewed as a vector space over the base field
mathematics, including algebra and number theory—indeed in any area where fields appear prominently. Suppose that E/F is a field extension. Then E may be considered
Degree_of_a_field_extension
Homological algebra is the study of homological functors
module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective
List of homological algebra topics
List_of_homological_algebra_topics
enveloping algebra of a Lie algebra. extension A ring E is a ring extension of a ring R if R is a subring of E. exterior algebra The exterior algebra of a vector
Glossary_of_ring_theory
248-dimensional exceptional simple Lie group
several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding
E8_(mathematics)
algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations. Vector algebra
Comparison of vector algebra and geometric algebra
Comparison_of_vector_algebra_and_geometric_algebra
In mathematics, a type of algebra
solvable, while a central extension of a nilpotent algebra by a nilpotent algebra is nilpotent. A solvable nonzero Lie algebra has a nonzero abelian ideal
Solvable_Lie_algebra
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