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Polynomial coprime with its derivative
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct
Separable_polynomial
Type of algebraic field extension
a separable extension if for every α ∈ E {\displaystyle \alpha \in E} , the minimal polynomial of α {\displaystyle \alpha } over F is a separable polynomial
Separable_extension
Mathematical expression for linear operators
every polynomial is a product of separable polynomials (since every polynomial is a product of its irreducible factors, and these are separable over a
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
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
Algebraic field extension
E/F} is a normal extension and a separable extension. E {\displaystyle E} is a splitting field of a separable polynomial with coefficients in F . {\displaystyle
Galois_extension
Topics referred to by the same term
Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation Separable polynomial, a polynomial whose
Separability
Topological space with a dense countable subset
In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle
Separable_space
Function of the coefficients of a polynomial that gives information on its roots
factor which is not separable (i.e., the irreducible factor is a polynomial in x p {\displaystyle x^{p}} ). The discriminant of a polynomial is, up to a scaling
Discriminant
Alebraic concept
\alpha \in E\setminus F} , the minimal polynomial of α {\displaystyle \alpha } over F is not a separable polynomial. If F is any field, the trivial extension
Purely_inseparable_extension
Every polynomial has a real or complex root
non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Error-correcting codes
code belongs to the class of maximum distance separable codes. While the number of different polynomials of degree less than k and the number of different
Reed–Solomon_error_correction
Algebraic structure
derivative. Every irreducible polynomial over K {\displaystyle K} is separable. Every finite extension of K {\displaystyle K} is separable. Every algebraic extension
Perfect_field
Polynomial sequence
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets
Hermite_polynomials
polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable
Separable_permutation
Polynomial sequence
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike
Zernike_polynomials
Field theory is the branch of algebra that studies fields
extension generated by roots of separable polynomials. Perfect field A field such that every finite extension is separable. All fields of characteristic
Glossary_of_field_theory
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
Field extension that is not algebraic
{\displaystyle K} ; that is, an element that is not a root of any univariate polynomial with coefficients in K {\displaystyle K} . In other words, a transcendental
Transcendental_extension
Topic in algebraic number theory
. A polynomial P ( x ) {\displaystyle P(x)} with coefficients in k {\displaystyle k} is called an additive polynomial, or a Frobenius polynomial, if P
Additive_polynomial
Mathematical connection between field theory and group theory
introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms
Galois_theory
Invariant of polynomial roots
group in degree five. It is a polynomial of degree 6. These three resolvents have the property of being always separable, which means that, if they have
Resolvent_(Galois_theory)
irreducible polynomial p ( x ) = ( x − a ) ∑ i = 0 n − 1 b i x i {\textstyle p(x)=(x-a)\sum _{i=0}^{n-1}b_{i}x^{i}} , then a separability idempotent is
Separable_algebra
Ring produced from two fields
over the finite 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
Tensor_product_of_fields
Theory in abstract algebra
because if α and β are roots of the cubic polynomial, we shall have (α/β)3 =1 and the cubic is a separable polynomial. Then L/K is a Kummer extension. More
Kummer_theory
Algebraic structure
satisfies the polynomial equation x p n − x = 0 {\displaystyle x^{p^{n}}-x=0} . Any finite field extension of a finite field is separable and simple. That
Finite_field
Mathematical group
describing the solutions to quintic polynomials. The study of field extensions and their relationship to the polynomials that give rise to them via Galois
Galois_group
Construction of a larger algebraic field by "adding elements" to a smaller field
L / K {\displaystyle L/K} is called separable if the minimal polynomial of every element of L over K is separable, i.e., has no repeated roots in an algebraic
Field_extension
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)| = [L:K]
Glossary_of_number_theory
Branch of mathematics that studies algebraic structures
Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial Normal extension Galois extension Abelian extension Transcendence
List of abstract algebra topics
List_of_abstract_algebra_topics
Algebraic structure with addition, multiplication, and division
E[X] / f(X), where f is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of f are contained
Field_(mathematics)
Field theory theorem
field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem
Primitive_element_theorem
Field generated by all rupture-fields of a polynomial over a field
(if we assume it is separable). A splitting field of a set P of polynomials is the smallest field over which each of the polynomials in P splits. An extension
Splitting_field
\mathbb {F} _{p}(t)} is non-separable, hence the associated morphism of schemes is not smooth. If we look at the minimal polynomial of the field extension
Smooth_morphism
Mathematical theorem in the study of analysis
theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by
Stone–Weierstrass_theorem
Code added to allow recovery of lost data
optimal reception efficiency). Optimal erasure codes are maximum distance separable codes (MDS codes). Parity check is the special case where n = k + 1. From
Erasure_code
Function used in signal processing
the coordinate axes. Only the Gaussian function is both separable and isotropic. The separable forms of all other window functions have corners that depend
Window_function
Partially unsolved problem in mathematics
specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, each example, found so far, of an operator
Invariant_subspace_problem
Complex-valued function
in modernized notation, that it can be expanded in terms of Hermite polynomials H ( ⋅ ) {\displaystyle H(\cdot )} based on weight function exp ( −
Mehler_kernel
Relates the topology of a complete non-archimedean field to its algebraic extensions
extensions. Let K be a complete non-archimedean field and let K be a separable closure of K. Given an element α in K, denote its Galois conjugates by
Krasner's_lemma
Topics referred to by the same term
extension by elements that do not all satisfy a separable polynomial Inseparable polynomial, a polynomial that does not have distinct roots in a splitting
Inseparable
Generalization of games used in game theory
written as a multivariate polynomial. In general, mixed Nash equilibria of separable games are easier to compute than non-separable games as implied by the
Continuous_game
Function with a multiplicative scaling behaviour
kth-degree or kth-order homogeneous function. For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above definition
Homogeneous_function
even on an uncountable supporting set, giving an example of non-separable polynomially reflexive Banach space. Distortion problem Sequence space, Schauder
Tsirelson_space
Graph whose induced subgraphs preserve distance
discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph
Distance-hereditary_graph
prime). Taking g=1 and f any ordinary separable polynomial shows that any differentially closed field is separably closed. In characteristic 0 this implies
Differentially_closed_field
On invertibility of polynomial maps (mathematics)
conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself
Jacobian_conjecture
Number representing a continuous quantity
are not rational are irrational. Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the
Real_number
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
matrices, polynomial-time decoding algorithms are known; the naïve algorithm is O ( n t ) {\displaystyle O(nt)} . For arbitrary d-separable but non-d-disjunct
Disjunct_matrix
Differential equation containing derivatives with respect to only one variable
differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the
Ordinary differential equation
Ordinary_differential_equation
Unsolved problem in number theory
problem about polynomials which have factors in common with their derivatives, proposed by Eduardo Casas-Alvero in 2001. Let f be a polynomial of degree d
Casas-Alvero_conjecture
Field extension generated by a one element
completely classified. The primitive element theorem states that every finite separable extension is a simple extension. For a field of characteristic 0 such
Simple_extension
Mathematical function
function fields; in the separable case, they may also be studied via finite or ramified covers of the projective line. Polynomial and rational functions
Algebraic_function
Roots of an algebraic element's minimal polynomial
algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K. Conjugate elements are commonly called conjugates
Conjugate element (field theory)
Conjugate_element_(field_theory)
1 {\displaystyle \mathbb {P} ^{1}} . Separability of defining polynomial g {\displaystyle g} ensures separability of the corresponding function field extension
Artin–Schreier_curve
Notion for convergence of metric spaces
sequence. The Gromov–Hausdorff space is path-connected, complete, and separable. It is also geodesic, i.e., any two of its points are the endpoints of
Gromov–Hausdorff_convergence
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
Class of games in game theory
CG with separable costs and resource-independent weights with eight players in which no PNE exists. When cost functions are additively-separable with linear
Congestion_game
Subpermutation of a longer permutation
{\displaystyle {\mathcal {C}}} -Pattern PPM and it was shown to be polynomial-time solvable for separable permutations. Later, Jelínek and Kynčl completely resolved
Permutation_pattern
Tool for solving polynomial equations
mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the
Newton_polygon
Differential form in commutative algebra
differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed
Kähler_differential
Theorem in algebraic number theory
\alpha \in {\mathcal {O}}_{K}} and f {\displaystyle f} be the minimal polynomial of α {\displaystyle \alpha } over Z [ x ] {\displaystyle \mathbb {Z} [x]}
Dedekind–Kummer_theorem
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
(B)\to \operatorname {Spec} (A)} is unramified if and only if the polynomial F is separable (i.e., it and its derivative generate the unit ideal of A [ t
Unramified_morphism
Algebraic structure with addition and multiplication
complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring
Ring_(mathematics)
Concept in algebraic geometry
{\displaystyle R} -algebra. Choose a monic polynomial f {\displaystyle f} in R [ x ] {\displaystyle R[x]} and a polynomial g {\displaystyle g} in R [ x ] {\displaystyle
Étale_morphism
Polygon intersected up to twice by lines orthogonal to a given line
line) can be performed in polynomial time. In the context of motion planning, two nonintersecting monotone polygons are separable by a single translation
Monotone_polygon
Mathematical arithmetic dynamics function
{\displaystyle K} be a field and K s e p {\displaystyle K^{sep}} be its separable closure. The Galois group G K {\displaystyle G_{K}} of the extension K
Arboreal Galois representation
Arboreal_Galois_representation
Mathematical operation
Galois theory, where the distinction is made between separable field extensions (defined by polynomials with no multiple roots) and inseparable ones. When
Formal_derivative
German mathematician (1882–1935)
the polynomial has no roots, because any choice of x makes the polynomial greater than or equal to one. If the field is extended then the polynomial may
Emmy_Noether
Concept in algebraic geometry
defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. The affine scheme X is smooth of dimension m over
Smooth_scheme
Swedish mathematician and concert pianist
Banach in his book, Theory of Linear Operators. Banach asked whether every separable Banach space has a Schauder basis. A Schauder basis or countable basis
Per_Enflo
Mathematical term in group theory
example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed
Grigorchuk_group
Monochrome light beam whose amplitude envelope is a Gaussian function
amplitude profiles are separable in x and y using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in r and θ using cylindrical
Gaussian_beam
Representation of a group or algebra that is a direct sum of simple representations
operator) if and only if the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials. Given a finite-dimensional representation
Semisimple_representation
between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K. Again, let L / K {\displaystyle
Finite extensions of local fields
Finite_extensions_of_local_fields
Generalization of covers
plane curve defined by the equation f(x,y) = 0, where f is a separable and irreducible polynomial in two indeterminates. If n is the degree of f in y, then
Branched_covering
Correspondence between subfields and subgroups
equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H {\displaystyle H} is a normal subgroup of
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
Algebraic curve
y^{2}+h(x)y=f(x)} where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the
Hyperelliptic_curve
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
two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within K a polynomial that does not always factorise. One
Thin_set_(Serre)
Extension of a mathematical field with polynomial roots
are a root of some nonzero polynomial with coefficients in k. Integral element Lüroth's theorem Galois extension Separable extension Normal extension
Algebraic_extension
Parity of a permutation Josephus permutation Parity of a permutation Separable permutation Stirling permutation Superpattern Transposition (mathematics)
List_of_permutation_topics
Local ring in which Hensel's lemma holds
Henselian local ring is called strictly Henselian if its residue field is separably closed. By abuse of terminology, a field K {\displaystyle K} with valuation
Henselian_ring
In mathematics, element that equals its square
idempotent a of R is called a full idempotent if RaR = R. A separability idempotent; see Separable algebra. Any non-trivial idempotent a is a zero divisor
Idempotent_(ring_theory)
to the field's Dedekind zeta function. Casas-Alvero conjecture: if a polynomial of degree d {\displaystyle d} defined over a field K {\displaystyle K}
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Cryptographic scheme
the secret value X {\displaystyle X} is a vector of many individually separable values. X = ( x 1 , x 2 , . . . , x n ) {\displaystyle X=(x_{1},x_{2}
Commitment_scheme
Mathematical element
said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of
Integral_element
Type of artificial neural network
stochastic gradient descent, which was able to classify non-linearily separable pattern classes. Amari's student Saito conducted the computer experiments
Feedforward_neural_network
) {\displaystyle G(A)} in polynomial time for several nonlinear objective functions including[citation needed]: Separable-convex functions of the form
Graver_basis
Set of methods for supervised statistical learning
space, it often happens that the sets to discriminate are not linearly separable in that space. For this reason, it was proposed that the original finite-dimensional
Support_vector_machine
When the utilities are SPLC (Separable Piecewise-Linear Concave) and either n or m is a constant, their algorithm is polynomial in the other parameter. The
Arrow–Debreu_exchange_market
Type of finite commutative rings
} is a root of the polynomial f ( x ) = x 3 + 2 x 2 + x − 1 {\displaystyle f(x)=x^{3}+2x^{2}+x-1} . Although any monic polynomial of degree 3 which is
Galois_ring
Russian mathematician (1935–2017)
1967, 293–298. V.I. Levenshtein, On the redundancy and deceleration of separable coding of natural numbers, Problems of Cybernetics, vol. 20, Nauka, Moscow
Vladimir_Levenshtein
non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases
Dual basis in a field extension
Dual_basis_in_a_field_extension
Subset whose closure is the whole space
uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C [ a , b ] {\displaystyle
Dense_set
by J, consisting of all elements x such that xJ⊆I. [] R[x,y,...] is a polynomial ring over R. [[]] R[[x,y,...]] is a formal power series ring over R. {}
Glossary of commutative algebra
Glossary_of_commutative_algebra
Vector space of infinite sequences
{\displaystyle \textstyle \ell ^{p}} and c 0 {\displaystyle c_{0}} are separable, with the sole exception of ℓ ∞ {\displaystyle \textstyle \ell ^{\infty
Sequence_space
Conjecture in graph theory
being connected. Tutte polynomial Characteristic polynomial Planarity The number of spanning trees in a graph Chromatic polynomial Being a perfect graph
Reconstruction_conjecture
Matrices similar to diagonal matrices
roots of unity on the diagonal. This follows since the minimal polynomial is separable, because the roots of unity are distinct. Projections are diagonalizable
Diagonalizable_matrix
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
Girl/Female
Muslim
Inseparable friend
Boy/Male
Muslim/Islamic
Inseparable friend
Boy/Male
Indian, Marathi
Separate
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Girl/Female
Indian, Punjabi, Sikh
Triumph of the Inseparable Creator
Girl/Female
Muslim
Example, Allegory, Parable
Girl/Female
Arabic, Muslim
Inseparable Friend
Surname or Lastname
English
English : variant spelling of Rimer 1.German : variant of Riemer.German : habitational name for someone from Riem (now a suburb of Munich; formerly a separate town).
Girl/Female
Biblical
A parable, governing.
Girl/Female
Muslim/Islamic
Inseparable friend
Biblical
a parable; governing
Girl/Female
Arabic, Muslim, Sindhi
Inseparable Friend
Girl/Female
Indian, Punjabi, Sikh
Love of the Inseparable Creator
Girl/Female
Indian
Inseparable
Surname or Lastname
English
English : occupational name for a maker of arms and armor, from Anglo-Norman French armer ‘arms-maker’ (Old French armier). Originally this was a separate name from Armour, but in due course the two became inextricably confused.
Girl/Female
Arabic
Separate
Boy/Male
Arabic, Australian, Muslim
Considerate; Inseparable Friend
Girl/Female
Arabic, Muslim
Example; Allegory; Parable
Boy/Male
Muslim
Considerate, Inseparable friend
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
Girl/Female
Tamil
Teekshika | திகà¯à®·à¯€à®•à®¾Â
Girl/Female
Muslim
Reformer, Advisor
Girl/Female
Hindu, Indian
Creation
Boy/Male
Tamil
Name of a Hindu month
Girl/Female
Indian
Light; Sunshine
Girl/Female
Indian
(Wife of Lord Vishnu)
Girl/Female
Arabic, Australian, British, Chinese, Danish, English, Finnish, French, German, Hawaiian, Hebrew, Italian, Muslim, Swedish, Teutonic
Form of Willamina; Will; Desire; Helmet; Protection; Resolute; Strong; Love; Will-helmet; Mother; Bitterness; Child of the Red Earth
Female
Irish
Variant spelling of Irish Gaelic Fionnghuala, FINNGUALA means "white shoulder."
Girl/Female
Spanish
From Briseis, the woman Achilles loved in Homer's Iliad.
Boy/Male
Hindu
Beloved
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
SEPARABLE POLYNOMIAL
a.
Not separable; incapable of being separated or disjoined.
a.
Invariably attached to some word, stem, or root; as, the inseparable particle un-.
a.
Able to speak.
a.
Capable of being, or proper to be , repaid; due; as, a loan repayable in ten days; services repayable in kind.
n.
See Sperable.
adv.
In an inseparable manner or condition; so as not to be separable.
adv.
In a reparable manner.
a.
Reparable.
a.
That may be secured.
a.
Separable.
p. a.
Disunited from the body; disembodied; as, a separate spirit; the separate state of souls.
a.
Capable of being repaired, restored to a sound or good state, or made good; restorable; as, a reparable injury.
n.
A kind of small nail used by shoemakers.
v. t.
To represent by parable.
a.
Inseparable.
a.
Capable of being spoken; fit to be spoken.
a.
Capable of being overcome or conquered; surmountable.
a.
Capable of being prepared.
a.
Capable of being separated, disjoined, disunited, or divided; as, the separable parts of plants; qualities not separable from the substance in which they exist.
a.
Capable of being severed.