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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
Surjective ring homomorphism with a given codomain
of Algebraic Schemes. Springer Science & Business Media. ISBN 978-3-540-30615-3. algebra extension at nLab infinitesimal extension at nLab Extension of
Algebra_extension
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
Algebraic structure where all polynomials have roots
{\displaystyle K} form an algebraically closed field called an algebraic closure of K . {\displaystyle K.} Given two algebraic closures of K {\displaystyle
Algebraically_closed_field
Construction of a larger algebraic field by "adding elements" to a smaller field
and 1/s are all algebraic. An algebraic extension L / K {\displaystyle L/K} is an extension such that every element of L is algebraic over K. Equivalently
Field_extension
Finite extension of the rationals
The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory
Algebraic_number_field
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
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
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
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 with addition, multiplication, and division
Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly
Field_(mathematics)
Group for which a given group is a normal subgroup
In Lie group theory, central extensions arise in connection with algebraic topology. Roughly speaking, central extensions of Lie groups by discrete groups
Group_extension
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
Type of complex number
1 + i {\displaystyle 1+i} is algebraic because it is a root of the polynomial x 4 + 4 {\displaystyle x^{4}+4} . Algebraic numbers include all integers
Algebraic_number
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
Concept in abstract algebra
mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero
Algebraic_element
Alebraic concept
general notion of radical extensions. An algebraic extension E ⊇ F {\displaystyle E\supseteq F} is a purely inseparable extension if and only if for every
Purely_inseparable_extension
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
Complex number that solves a monic polynomial with integer coefficients
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root
Algebraic_integer
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
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
Mathematical function
these by composition and algebraic operations (addition, multiplication, subtraction, and division). Thus an example of an algebraic function is the function
Algebraic_function
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
Topics referred to by the same term
Look up algebraic in Wiktionary, the free dictionary. Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic
Algebraic
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
Every polynomial has a real or complex root
due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap. The other one was published
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Topics referred to by the same term
extension property, in topology Kolmogorov extension theorem, in probability theory Linear extension, in order theory Sheaf extension, in algebraic geometry
Extension
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
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
Algebra associated to any vector space
universal algebra. This then paved the way for the 20th-century developments of abstract algebra by placing the axiomatic notion of an algebraic system on
Exterior_algebra
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
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
Branching out of a mathematical structure
the example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one. Ramification in algebraic number theory means
Ramification_(mathematics)
Finitely generated extension field of positive transcendence degree
field extension K / k {\displaystyle K/k} which has transcendence degree n {\displaystyle n} over k {\displaystyle k} . Equivalently, an algebraic function
Algebraic_function_field
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
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
Set without nontrivial polynomial equalities
field extension L / K {\displaystyle L/K} that is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent
Algebraic_independence
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
Branch of mathematics that studies algebraic structures
theory) Field extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial
List of abstract algebra topics
List_of_abstract_algebra_topics
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
{\displaystyle {\mathcal {M}}} is continuous in the uniform topology. The algebraic extension of Δ {\displaystyle \Delta } assigns a value of 0 to a singular operator
Fuglede−Kadison_determinant
Branch of mathematics
empirical sciences. Algebra is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure is a non-empty
Algebra
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
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
Mathematical structure in abstract algebra
Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear
*-algebra
Mathematical operation on vector spaces
product over the base field is again algebraic over the base field. Specifically, it is the algebraic extension generated by the products of the generators
Tensor_product
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
Construction in homological algebra
homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and
Ext_functor
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
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
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
Specific algebraic group
commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled
Algebraic_torus
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Particular kind of algebraic structure
are singular in the given algebra A {\displaystyle A} have a multiplicative inverse element in a Banach algebra extension B . {\displaystyle B.} Topological
Banach_algebra
Field in mathematics similar to the real numbers
F is not algebraically closed, but its algebraic closure is a finite extension of F. F is not algebraically closed but the field extension F ( − 1 )
Real_closed_field
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
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
Algebra based on a vector space with a quadratic form
Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots
Clifford_algebra
Algebraic structure
{F} }}_{p}} be an algebraic closure of F p {\displaystyle \mathbb {F} _{p}} . It is unique up to isomorphism, as holds for an algebraic closure of any given
Finite_field
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
Type of Kac–Moody algebras
Lie algebra, the corresponding affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak {g}}}} is constructed as a central extension of the loop algebra g
Affine_Lie_algebra
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k. Eichler
Approximation in algebraic groups
Approximation_in_algebraic_groups
Branch of mathematics
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts
Derived_algebraic_geometry
Algebraic structure used in analysis
in algebraic terms. The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R. Namely, a Lie algebra g {\displaystyle
Lie_algebra
A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed. Any algebraic extension of a quasi-algebraically closed field
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Algebraic structure used in logic
Heyting algebras serve as the algebraic models of propositional intuitionistic logic in the same way Boolean algebras model propositional classical logic
Heyting_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)
Branch of algebraic number theory concerned with abelian extensions
(CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using
Class_field_theory
Method for computing the relation of two integers with their greatest common divisor
algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non-prime order. It follows
Extended_Euclidean_algorithm
Study of differential field extensions induced by linear differential equations
(2011): Algebraic Groups and Differential Galois Theory, AMS (GSM122), ISBN 978-0-8218-5318-4. Kovacic, J. J. (2005), Picard–Vessiot theory, algebraic groups
Picard–Vessiot_theory
Branch of number theory
Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields
Algebraic_number_theory
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
Greatest common divisor of polynomials
algorithm is that it allows one to compute division in algebraic field extensions. Let L an algebraic extension of a field K, generated by an element whose minimal
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Theory in abstract algebra
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots
Kummer_theory
Polynomial without nontrivial factorization
appear naturally in the study of polynomial factorization and algebraic field extensions. It is helpful to compare irreducible polynomials to prime numbers:
Irreducible_polynomial
Measure of a mathematical object studied in the field of algebraic geometry
are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
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
Branch of algebra that studies commutative rings
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings
Commutative_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
Roots of an algebraic element's minimal polynomial
field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial
Conjugate element (field theory)
Conjugate_element_(field_theory)
Algebraic manipulation of "true" and "false"
connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other
Boolean_algebra
Type of algebra
reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this
Finitely_generated_algebra
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 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 variety with a group structure
mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus
Algebraic_group
Reasoning about equations with free variables
logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses
Algebraic_logic
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
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
Topics referred to by the same term
the subset. In field theory, the normal closure of an algebraic extension F/K is an extension field L of F such that L/K is normal and L is minimal with
Normal_closure
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
Second homology group of a group
central extension, but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of
Schur_multiplier
Generalization of algebraic variety
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety in several ways, such as taking
Scheme_(mathematics)
Property in algebraic topology
In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to
Homotopy_extension_property
Mathematical connection between field theory and group theory
field. It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses
Galois_theory
Generalization of covers
on each circle. In algebraic geometry, the term branched covering is used to describe morphisms f {\displaystyle f} from an algebraic variety V {\displaystyle
Branched_covering
Generalized natural number
encode the algebraic extensions of a finite field. Supernatural numbers also arise in the classification of uniformly hyperfinite algebras. Profinite
Supernatural_number
Study of systems of inequalitites
mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with
Real_algebraic_geometry
Computer algebra system
Algebraic Computation (SIGSAM '89). ACM. pp. 207–211. Claire Dicrescenzo; Dominique Duval (1989). P. Gianni (ed.). Algebraic extensions and algebraic
Axiom (computer algebra system)
Axiom_(computer_algebra_system)
Number system extending the rational numbers
proper algebraic extension: the complex numbers C {\displaystyle \mathbb {C} } . In other words, this quadratic extension is already algebraically closed
P-adic_number
Mathematical concept in algebraic geometry
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
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