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Type of algebraic field extension
a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle \alpha
Separable_extension
Algebraic field extension
a separable closure of K {\displaystyle K} . Since a separable extension of a separable extension is again separable, there are no finite separable extensions
Algebraic_closure
Topics referred to by the same term
of a separable field extension Separable differential equation, in which separation of variables is achieved by various means Separable extension, in field
Separability
Algebraic field extension
In 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
Polynomial coprime with its derivative
Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic
Separable_polynomial
a 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
Sufficient condition for a separable extension of a Hilbertian field to be Hilbertian
condition for a separable extension of a Hilbertian field to be Hilbertian. Let K be a Hilbertian field and L a separable extension of K. Assume there
Haran's_diamond_theorem
Construction of a larger algebraic field by "adding elements" to a smaller field
extension is a field extension that is both normal and separable. A consequence of the primitive element theorem states that every finite separable extension
Field_extension
Field extension generated by a one element
every finite separable extension is a simple extension. For a field of characteristic 0 such as the rationals, this means all finite extensions are simple
Simple_extension
Galois extension whose Galois group is abelian
that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements
Abelian_extension
Field extension that is not algebraic
transcendence basis S such that L is a separable algebraic extension over K(S). A field extension L / K is said to be separably generated if it admits a separating
Transcendental_extension
Extension of a mathematical field with polynomial roots
coefficients in k. Integral element Lüroth's theorem Galois extension Separable extension Normal extension Fraleigh (2014), Definition 31.1, p. 283. Malik, Mordeson
Algebraic_extension
of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K. Again, let L / K
Finite extensions of local fields
Finite_extensions_of_local_fields
Specific algebraic group
field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of Gm/L. In general,
Algebraic_torus
Alebraic concept
of α {\displaystyle \alpha } over F is not a separable polynomial. If F is any field, the trivial extension F ⊇ F {\displaystyle F\supseteq F} is purely
Purely_inseparable_extension
Field theory is the branch of algebra that studies fields
Separable extension An extension generated by roots of separable polynomials. Perfect field A field such that every finite extension is separable. All
Glossary_of_field_theory
Mathematical element
integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure A ′ {\displaystyle A'} of A in L is a
Integral_element
Type of vector space in math
Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if
Hilbert_space
Algebraic structure with addition, multiplication, and division
extensions F / E, which are, by definition, those that are separable and normal. The primitive element theorem shows that finite separable extensions
Field_(mathematics)
Field theory theorem
theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies
Primitive_element_theorem
Legal concept regulating rights of a creative work
the United States was increased by 20 years under the Copyright Term Extension Act. This legislation was the subject of substantial criticism following
Copyright
to a finite product of finite separable field extensions. An étale algebra is a special sort of commutative separable algebra. Let K be a field. Let
Étale_algebra
Algebraic structure with addition and multiplication
is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension A ⊗ k F {\displaystyle
Ring_(mathematics)
Concept in algebraic geometry
is a finite extension field of k. The variety X is smooth over k if and only if E is a separable extension of k. Thus, if E is not separable over k, then
Smooth_scheme
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 K
Perfect_field
Algebraic structure
x = 0 {\displaystyle x^{p^{n}}-x=0} . Any finite field extension of a finite field is separable and simple. That is, if E {\displaystyle E} is a finite
Finite_field
Correspondence between subfields and subgroups
a normal extension of F {\displaystyle F} (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field. Special groups were defined in 1958
Special group (algebraic group theory)
Special_group_(algebraic_group_theory)
general non-separable Banach spaces. A slightly modified version also commonly referred to as the Sobczyk theorem, deals with the extension of a bounded
Sobczyk's_theorem
disjoint from the separable closure of K over K. A subextension of a primary extension is primary. A primary extension of a primary extension is primary (transitivity)
Primary_extension
Field generated by all rupture-fields of a polynomial over a field
reasoning. Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′
Splitting_field
Subject area in mathematics
{\displaystyle \mu _{m}} denotes the group of m-th roots of unity in some separable extension of k. This extends to ∂ n : k ∗ × ⋯ × k ∗ → H n ( k , μ m ⊗ n )
Algebraic_K-theory
Mathematical group
Galois extensions E / F {\displaystyle E/F} for a fixed field. The inverse limit is denoted Gal ( F ¯ / F ) := lim ← E / F finite separable Gal
Galois_group
Type of field extension
closed field is regular. An extension is regular if and only if it is separable and primary. A purely transcendental extension of a field is regular. There
Regular_extension
Local ring in which Hensel's lemma holds
finite ring extension is a product of local rings. A Henselian local ring is called strictly Henselian if its residue field is separably closed. By abuse
Henselian_ring
{\displaystyle R} is an integral domain and L {\displaystyle L} a finite separable extension of K {\displaystyle K} , then the integral closure S {\displaystyle
Order_(ring_theory)
Branch of mathematics that studies algebraic structures
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
empty set is a p-basis, though the extension is separable and has transcendence degree 1. If K is a degree p extension of k obtained by adjoining a pth
P-basis
applications the category D is sometimes the category of models of finite separable extensions of a global field, and Φ corresponds to the line bundles on these
Frobenioid
Mathematical terminology
field theory, the union of the idele class groups of all finite separable extensions of K is used instead. There are also Galois representations that
Galois_representation
Finite dimensional algebra over a field whose central elements are that field
theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of K of degree equal to the index of A, and this splitting field
Central_simple_algebra
Ring produced from two fields
field with p elements (see Separable polynomial: the point here is that P is not separable). If L is the field extension K(T 1/p) (the splitting field
Tensor_product_of_fields
Theory in abstract algebra
polynomial, we shall have (α/β)3 =1 and the cubic is a separable polynomial. Then L/K is a Kummer extension. More generally, it is true that when K contains
Kummer_theory
Theorem on extension of bounded linear functionals
analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector
Hahn–Banach_theorem
Book about number theory
Tensor products are used to study extensions of the places of an A-field to places of a finite separable extension of the field, with the more complicated
Basic_Number_Theory
Algorithm for supervised learning of binary classifiers
linear separability of the training set is not known a priori, one of the training variants below should be used. Detailed analysis and extensions to the
Perceptron
About polynomials in several variables
Mathematica Vietnamica, 32: 205–214, MR 2368008 Adjamagbo, Kossivi (1995), "On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic"
Jacobian_conjecture
Structure in algebraic geometry
motivic Galois group, fix a field k and consider the functor finite separable extensions K of k → non-empty finite sets with a (continuous) transitive action
Motive_(algebraic_geometry)
with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let I A
Ideal_norm
Theorem in measure theory
Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later
Prokhorov's_theorem
Function in algebra
When L/K is separable, the ramification index of w over v is defined to be e(w/v)pi, where pi is the inseparable degree of the extension Rw/mw over Rv/mv
Valuation_(algebra)
Image edge detection algorithm
Sobel–Feldman operator is based on convolving the image with a small, separable, and integer-valued filter in the horizontal and vertical directions and
Sobel_operator
Type of morphism in algebraic geometry
local ring of x in the fiber f−1(f(x)), is a field and a finite separable extension of κ(f(x)), then f is unramified at x. Finite morphisms are quasi-finite
Quasi-finite_morphism
Mathematical expression for linear operators
polynomial of x {\displaystyle x} is a product of separable polynomials, then the field extension L / K {\displaystyle L/K} is Galois, meaning that F
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Galois group of the separable closure
{\displaystyle K} , where K sep {\displaystyle K^{\textrm {sep}}} is a separable closure of K {\displaystyle K} . Alternatively, it is the group of all
Absolute_Galois_group
Collection of random variables
the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob’s separability theorem
Stochastic_process
Concept in topology
the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic
Polish_space
Kadison, L.; Nikshych, D. (2001), "Hopf algebra actions of strongly separable extensions of depth two", Advances in Mathematics, 163 (2): 258–286, arXiv:math/0107064
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
are satisfied: L/K is a normal extension and a separable extension, L is a splitting field of a separable polynomial with coefficients in K, |Aut(L/K)|
Glossary_of_number_theory
Theory in mathematics
generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi
KK-theory
Math concept
{\displaystyle A} , then the above homotopy can be taken to be an extension of that. Conversely, a separable metric space Y {\displaystyle Y} is an ANR if there exists
Absolute_neighborhood_retract
O(l)=O(k)\times _{X}O(l)} Recall that a field extension K → L {\displaystyle K\to L} is called separable iff given a presentation L = K [ x ] ( f ( x )
Smooth_morphism
Hilbertianity is preserved under finite separable extensions and abelian extensions. If N is a Galois extension of a Hilbertian field, then although N
Thin_set_(Serre)
Differential form in commutative algebra
a finite field extension, then Ω K / k 1 = 0 {\displaystyle \Omega _{K/k}^{1}=0} if and only if K / k {\displaystyle K/k} is separable. Consequently,
Kähler_differential
Relates the topology of a complete non-archimedean field to its algebraic extensions
non-archimedean field to its algebraic extensions. Let K be a complete non-archimedean field and let K be a separable closure of K. Given an element α in
Krasner's_lemma
Mathematical function
_{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).} If L/K is separable then each root appears only once (however this does not mean the coefficient
Field_trace
restriction of G m {\displaystyle \mathbb {G} _{\text{m}}} along a finite separable extension ℓ {\displaystyle \ell } of k {\displaystyle k} , so that T ( k )
Moy–Prasad_filtration
Effort being used in the working memory
(2018). "A comparison of three measures of cognitive load: Evidence for separable measures of intrinsic, extraneous, and germane load". IEEE Transactions
Cognitive_load
Normed vector space that is complete
of a separable Banach space need not be separable, but: Theorem—Let X {\displaystyle X} be a normed space. If X ′ {\displaystyle X'} is separable, then
Banach_space
Group of unitary matrices
the field extension can be replaced by any degree 2 {\displaystyle 2} separable algebra, most notably a degree 2 {\displaystyle 2} extension of a finite
Unitary_group
Group controlling representation theory
the full Galois group Gal(K/k) of the separable closure, one can just use the Galois group of a finite extension over which G is split. The corresponding
Langlands_dual_group
Ring that is also a vector space or a module
with the action x ⋅ (a ⊗ b) = axb. Then, by definition, A is said to separable if the multiplication map A ⊗R A → A : x ⊗ y ↦ xy splits as an Ae-linear
Associative_algebra
Vector space of functions in mathematics
, W k , p ( Ω ) {\displaystyle p<\infty ,W^{k,p}(\Omega )} is also a separable space. It is conventional to denote W k , 2 ( Ω ) {\displaystyle W^{k
Sobolev_space
Solving multiple machine learning tasks at the same time
represented solely by A. Methods for non-separable kernels Γ is a current field of research. For the separable case, the representation theorem is reduced
Multi-task_learning
Special group in linear algebra
These groups do not always agree. For example, let L be a finite separable extension of K of ramification degree e. The torus L×/K× is compact. However
Iwahori_subgroup
Concept in field theory mathematics
_{j=1}^{n}\sigma _{j}(\alpha ){\biggr )}^{[L:K(\alpha )]}} . If L/K is separable, then each root appears only once in the product (though the exponent
Field_norm
provides a non-degenerate quadratic form over K. This is true if L is separable over K; it is always true if K is a perfect field, including when K is
Dual basis in a field extension
Dual_basis_in_a_field_extension
Algebraic geometry
are formally étale. Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale
Formally_étale_morphism
Mathematical connection between field theory and group theory
theory only considers Galois extensions, which are in particular separable. General field extensions can be split into a separable, followed by a purely inseparable
Galois_theory
German mathematician (1871–1928)
transcendence degree of a field extension, and also normal and separable extensions (the latter he called algebraic extensions of the first kind). Besides
Ernst_Steinitz
Restriction of scalars
} if L is separable over k. Restriction of scalars on abelian varieties (e.g. elliptic curves) yields abelian varieties, if L is separable over k. James
Weil_restriction
Complex-differentiable (mathematical) function
Biholomorphy Cauchy's estimate Harmonic maps Harmonic morphisms Holomorphic separability Meromorphic function Quadrature domains Wirtinger derivatives "Analytic
Holomorphic_function
Algebraic structure
situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case
Tate_module
{\displaystyle s^{3}=y} . separable An algebra over a field is called separable if its extension by any finite purely inseparable extension is reduced. separated
Glossary of commutative algebra
Glossary_of_commutative_algebra
Graphic-art effect
implementing the box blur of a radius r and N pixels: The box blur is a separable filter, so that only two 1D passes of averaging 2 r + 1 pixels will be
Box_blur
Algebraic structure with "nice" duality properties
example of a separable algebra extension since e = ∑ i = 1 n a i ⊗ B b i {\textstyle e=\sum _{i=1}^{n}a_{i}\otimes _{B}b_{i}} is a separability element satisfying
Frobenius_algebra
Concept in mathematics
-almost everywhere to a function g {\displaystyle g} taking values in a separable subspace B 0 {\displaystyle B_{0}} of B {\displaystyle B} , and such that
Bochner_integral
irreducible for a separable closure k s {\displaystyle k_{s}} of k. X × k F {\displaystyle X\times _{k}F} is irreducible for each field extension F of k. The
Geometrically (algebraic geometry)
Geometrically_(algebraic_geometry)
Topological complex vector space
or to determine the extent of which classification is possible, for separable simple nuclear C*-algebras. We begin with the abstract characterization
C*-algebra
Locally compact topological group with an invariant averaging operation
Existence of left-invariant states. There is a left-invariant state on any separable left-invariant unital C*-subalgebra of the bounded continuous functions
Amenable_group
Evolutionary algorithm
applications and is considered to be particularly useful on non-convex, non-separable, ill-conditioned, multi-modal or noisy objective functions. One survey
CMA-ES
Roots of an algebraic element's minimal polynomial
in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)]sep. A theorem of Kronecker states that if α is a nonzero
Conjugate element (field theory)
Conjugate_element_(field_theory)
Quantum mechanical statistic
particles is separable, then the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Exact separability is extremely
Quantum_potential
Topological space
topology. Thus, the Moore plane shows that a subspace of a separable space need not be separable. The Moore plane is first countable, but not second countable
Moore_plane
American basketball player (born 1995)
at the Wayback Machine, Star Tribune, June 26, 2015 "Towns' parents separable, if only for UK games". courier-journal.com. Retrieved March 28, 2015
Karl-Anthony_Towns
Square matrix constructed from a monic polynomial
= C ( p ) . {\displaystyle [m_{\lambda }]=C(p).} Assuming this extension is separable (for example if F {\displaystyle F} has characteristic zero or is
Companion_matrix
Group comohology of Galois modules
quaternion algebras for instance). When the extension field L = K s {\displaystyle L=K^{s}} is the separable closure of the field K {\displaystyle K} ,
Galois_cohomology
have that The residue field k ( x ) {\displaystyle k(x)} is a separable algebraic extension of k ( y ) {\displaystyle k(y)} . f # ( m y ) O x , X = m x
Unramified_morphism
Statistical test for causality
purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems
Convergent_cross_mapping
*-algebra of bounded operators on a Hilbert space
Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. The von
Von_Neumann_algebra
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