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Polynomial ideals are finitely generated
mathematics, Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology)
Hilbert's_basis_theorem
Topics referred to by the same term
Basis theorem can refer to: Basis theorem (computability), a type of theorem in computability theory showing that sets from particular classes must have
Basis_theorem
Result about when a matrix can be diagonalized
spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This
Spectral_theorem
The low basis theorem is one of several basis theorems in computability theory, each of which show that, given an infinite subtree of the binary tree 2
Low_basis_theorem
On polynomial rings over fields
invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem, which asserts that all ideals of
Hilbert's_syzygy_theorem
Topics referred to by the same term
polynomial function of these basis elements Orthonormal basis of a Hilbert space Hilbert basis (linear programming) Hilbert's basis theorem This disambiguation
Hilbert_basis
for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study
Normal_basis
In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are
Basis_theorem_(computability)
Topics referred to by the same term
Gröbner basis Hilbert's basis theorem Generating set of a group Base (topology) Change of basis Greedoid Normal basis Polynomial basis Radial basis function
Basis
In linear algebra, relation between 3 dimensions
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity
Rank–nullity_theorem
Mathematical construct in computer algebra
polynomial rings are Noetherian (Hilbert's basis theorem). Condition 4 ensures that the result is a Gröbner basis, and the definitions of S-polynomials and
Gröbner_basis
Commutative group where every element is the sum of elements from one finite subset
represent G as such a decomposition. The proof of this statement uses the basis theorem for finite abelian group: every finite abelian group is a direct sum
Finitely generated abelian group
Finitely_generated_abelian_group
In mathematics, a statement that has been proven
theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements
Theorem
A prime p divides a^p–a for any integer a
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after
Fermat's_little_theorem
Mathematical ring with well-behaved ideals
with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem. For noncommutative rings,
Noetherian_ring
German mathematician (1862–1943)
a completely different path. As a result, he demonstrated Hilbert's basis theorem, showing the existence of a finite set of generators, for the invariants
David_Hilbert
Theorem in vector calculus
theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem,
Stokes'_theorem
Collection of subsets that generate a topology
Tychonoff's theorem, which states that the product of non-empty compact spaces is compact, has a short proof if the Alexander Subbase Theorem is used. Base
Subbase
Set of vectors used to define coordinates
which is called the dimension of V. This is the dimension theorem. A generating set S is a basis of V if and only if it is minimal, that is, no proper subset
Basis_(linear_algebra)
Branch of mathematical logic
are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast
Reverse_mathematics
Topics referred to by the same term
{R} ^{3}} Hilbert's Theorem 90, an important result on cyclic extensions of fields that leads to Kummer theory Hilbert's basis theorem, in commutative algebra
Hilbert's_theorem
Mathematical study of invariants under symmetries
Hilbert's basis theorem. Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on
Invariant_theory
Commutative group (mathematics)
abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic). The center Z ( G ) {\displaystyle Z(G)}
Abelian_group
Mathematical rule for inverting probabilities
formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". Stephen
Bayes'_theorem
Basic result in harmonic analysis on compact topological groups
} The final statement of the Peter–Weyl theorem (Knapp 1986, Theorem 1.12) gives an explicit orthonormal basis of L 2 ( G ) {\displaystyle L^{2}(G)} .
Peter–Weyl_theorem
free associative algebra generated by X. By the Poincaré–Birkhoff–Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra
Free_Lie_algebra
Theorem in dimensional analysis
Buckingham π theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states
Buckingham_pi_theorem
Theorem that any three objects in space can be simultaneously bisected by a plane
offers a proof of the theorem. A more modern reference is Stone & Tukey (1942), which is the basis of the name "Stone–Tukey theorem". This paper proves
Ham_sandwich_theorem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal
Additive_basis
All bases of a vector space have equally many elements
Formally, the dimension theorem for vector spaces states that: Given a vector space V, any two bases have the same cardinality. As a basis is a generating set
Dimension theorem for vector spaces
Dimension_theorem_for_vector_spaces
Explicitly describes the universal enveloping algebra of a Lie algebra
into the universal enveloping algebra U(L). Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite
Poincaré–Birkhoff–Witt theorem
Poincaré–Birkhoff–Witt_theorem
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to
Minkowski's_theorem
Computational tool
orthonormal basis of V. Let {bn} be a Schauder basis of a Banach space V over F = R or C. It is a subtle consequence of the open mapping theorem that the
Schauder_basis
Local theory of several complex variables
Rückert basis theorem. There is a deeper preparation theorem for smooth functions, due to Bernard Malgrange, called the Malgrange preparation theorem. It
Weierstrass preparation theorem
Weierstrass_preparation_theorem
Relation between algebraic varieties and polynomial ideals
1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem) and became a foundational result of algebraic geometry. There are several
Hilbert's_Nullstellensatz
Result in probability theory
characteristic functions. This theorem is the basis for one approach to prove the central limit theorem and is one of the major theorems concerning characteristic
Lévy's_continuity_theorem
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Topics referred to by the same term
Hilbert space dimension Hilbert dimension in ring theory, see Hilbert's basis theorem Hilbert series and Hilbert polynomial This disambiguation page lists
Hilbert_dimension
may be seen as a special case of Hilbert's basis theorem stating that every polynomial ideal has a finite basis, for the ideals generated by monomials. Indeed
Dickson's_lemma
Mathematical terminology
structure is. This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]-module of rank 1. If the same is true
Galois_representation
Theorem in quantum mechanics
In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from
Gleason's_theorem
Branch of mathematics
variables has a basis. He extended this further in 1890 to Hilbert's basis theorem. Once these theories had been developed, it was still several decades
Abstract_algebra
Branch of mathematics
prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connection between algebraic geometry
Algebraic_geometry
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians
Artin–Rees_lemma
Sufficiency theorem for reconstructing signals from samples
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
In additive number theory, a way to measure how dense a sequence of numbers is
an additive basis, and the least number of summands required is called the degree (sometimes order) of the basis. Thus, the last theorem states that any
Schnirelmann_density
Mathematical theorem
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum
Mercer's_theorem
Abstract algebra module
finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial
Noetherian_module
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Theorem in economics
tool in predicting possible economic outcomes. The Coase theorem is considered an important basis for most modern economic analyses of government regulation
Coase_theorem
Theorem constraining types of hidden-variable theories
quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon
Kochen–Specker_theorem
Branch of algebra that studies commutative rings
that polynomial rings over a field are Noetherian is called Hilbert's basis theorem. Moreover, many ring constructions preserve the Noetherian property
Commutative_algebra
Algebraic expansion of powers of a binomial
algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ( x
Binomial_theorem
Branch of functional analysis
functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups: Theorem— If A is a self-adjoint operator, then U t = e i t
Borel_functional_calculus
Theorem in mathematics
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following
Projection-slice_theorem
Economic theory about capital structure
The Modigliani–Miller theorem (of Franco Modigliani, Merton Miller) is an influential element of economic theory; it forms the basis for modern thinking
Modigliani–Miller_theorem
Mathematical result on infinite trees
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives
Kőnig's_lemma
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
On when a family of real, continuous functions has a uniformly convergent subsequence
family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary
Arzelà–Ascoli_theorem
Fundamental theorem in condensed matter physics
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves
Bloch's_theorem
Representation theory of groups
content of the normal basis theorem, a normal basis being an element x of L such that the g(x) for g in G are a vector space basis for L over K. Such x
Regular_representation
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed
Rayleigh theorem for eigenvalues
Rayleigh_theorem_for_eigenvalues
Field extension that is not algebraic
transcendence basis, L is algebraic over K(S); since L is also algebraically closed, it is an algebraic closure of K(S). The extension theorem therefore extends
Transcendental_extension
Hilbert's axioms Hilbert's basis theorem Hilbert's epsilon calculus Hilbert's inequality Hilbert's irreducibility theorem Hilbert's lemma Hilbert's Nullstellensatz
List of things named after David Hilbert
List_of_things_named_after_David_Hilbert
Mathematical theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of
Riesz–Fischer_theorem
Theorem in calculus relating line and double integrals
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R
Green's_theorem
Group theory theorem
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is
Closed-subgroup_theorem
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f
Brouwer_fixed-point_theorem
Hilbert's basis theorem Hilbert's axioms Hilbert function Hilbert's irreducibility theorem Hilbert's syzygy theorem Hilbert's Theorem 90 Hilbert's theorem Mathematics
List of scientific laws named after people
List_of_scientific_laws_named_after_people
American mathematician
faculty since 1967. He proved, together with Carl Jockusch, the low basis theorem, and has done other work in mathematical logic, primarily in the area
Robert_I._Soare
Statement in abstract algebra
algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Algorithm for computing Gröbner bases
the leading terms of our set F, and Dickson's lemma (or the Hilbert basis theorem) guarantees that any such ascending chain must eventually become constant
Buchberger's_algorithm
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Theorem in transcendental number theory
Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem—if α1
Lindemann–Weierstrass_theorem
Statistical theorem in the analysis of variance
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics
Cochran's_theorem
Type of vector space in math
space (the cardinality of a Hamel basis). Koashi, Masato, "Appendix: Linear algebra" (PDF) Hewitt & Stromberg (1965, Theorem 16.29) Prugovečki 1981, I, §4
Hilbert_space
Formula for area of a grid polygon
direction, using Pick's theorem (proved in a different way) as the basis for a proof of Euler's formula. Alternative proofs of Pick's theorem that do not use
Pick's_theorem
Every positive integer is a sum of at most n n-gonal numbers
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every
Fermat polygonal number theorem
Fermat_polygonal_number_theorem
Branch of mathematics that studies algebraic structures
basis theorem Hopkins–Levitzki theorem Krull's principal ideal theorem Levitzky's theorem Galois theory Abel–Ruffini theorem Wedderburn–Artin theorem
List of abstract algebra topics
List_of_abstract_algebra_topics
Concept in number theory
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers α
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Group comohology of Galois modules
algebraic number theory and the arithmetic of elliptic curves. The normal basis theorem implies that the first cohomology group of the additive group of L will
Galois_cohomology
Theorem in physics
Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with
Bell's_theorem
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Mathematical theorem
In mathematics, the transfinite recursion theorem says a function can be defined using a recursion over a well-ordered set; for example, N {\displaystyle
Transfinite_recursion_theorem
Theorem in abstract algebra
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
Method of proof in mathematics
considered problems seems to be Hilbert's Nullstellensatz and Hilbert's basis theorem. From a philosophical point of view, the former is especially interesting
Constructive_proof
Characterizes when a topological space is metrizable
topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space X {\displaystyle
Nagata–Smirnov metrization theorem
Nagata–Smirnov_metrization_theorem
Theory of stochastic processes
orthonormal basis of L2([a, b]) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
Results on the surface areas and volumes of surfaces and solids of revolution
Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with
Pappus's_centroid_theorem
Area of mathematics
Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered
Functional_analysis
Every Boolean algebra is isomorphic to a certain field of sets
Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to
Stone's representation theorem for Boolean algebras
Stone's_representation_theorem_for_Boolean_algebras
Product of any collection of compact topological spaces is compact
Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named
Tychonoff's_theorem
Ring in abstract algebra
numbers n. In contrast, if R is Noetherian so is R[x] by the Hilbert basis theorem. The ring of integers Z {\displaystyle \mathbb {Z} } is a Noetherian
Artinian_ring
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
Algebraic curve in mathematics
Silverman 1986, Theorem 4.1 Silverman 1986, pp. 199–205 See also Cassels, J. W. S. (1986). "Mordell's Finite Basis Theorem Revisited". Mathematical
Elliptic_curve
Mathematical proposition equivalent to the axiom of choice
several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's
Zorn's_lemma
Foundational result in symplectic geometry
Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It
Darboux's_theorem
BASIS THEOREM
BASIS THEOREM
Male
English
 English form of French Basile, BASIL means "king." Also sometimes given as an herb name.
Boy/Male
Tamil
King, Basil the herb
Boy/Male
Muslim
Smiling, Happy
Boy/Male
Turkish
Intelligent.
Surname or Lastname
English and French
English and French : from a medieval personal name, ultimately from Greek Basileios ‘royal’. The name was borne by a 4th-century bishop of Caesarea in Cappadocia, regarded as one of the four Fathers of the Eastern Church; he wrote important theological works and established a rule for religious orders of monks. Various other saints are also known under these and cognate names. The popularity of Vasili as a Russian personal name is largely due to the fact that this was the ecclesiastical name of St. Vladimir (956–1015), Prince of Kiev, who was chiefly responsible for the introduction of Christianity to Russia. As an American surname, this has also absorbed some Greek, Russian, and other derivatives of Greek Vasili.
Boy/Male
Muslim
King, Basil the herb (1)
Boy/Male
Muslim
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Boy/Male
Greek
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Boy/Male
Indian
Smiling, Happy
Boy/Male
Muslim Arabic
Smiling.
Boy/Male
Muslim
Vast, Spacious, One who stretches, Enlarges
Surname or Lastname
English
English : variant of Bayliss.Hungarian and Croatian (Bališ) : from the personal name Bali, a pet form of Baltazar or Balint.Perhaps also Greek : occupational status name from Turkish balija ‘workman’, ‘low-ranking man’.
Surname or Lastname
English
English : probably a variant spelling of Bevis.
Boy/Male
Muslim
Clear
Boy/Male
Greek American English
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Female
Hebrew
 Variant spelling of Hebrew Basya, BASIA means "daughter of God."
Boy/Male
Hindu
King, Basil the herb
Surname or Lastname
English
English : from Old French bas(se) ‘low’, ‘short’ (Latin bassus ‘thickset’; see Basso), either a descriptive nickname for a short person or a status name meaning ‘of humble origin’, not necessarily with derogatory connotations.English : in some instances, from Middle English bace ‘bass’ (the fish), hence a nickname for a person supposedly resembling this fish, or a metonymic occupational name for a fish seller or fisherman.Scottish : habitational name from a place in Aberdeenshire, of uncertain origin.Jewish (Ashkenazic) : metonymic occupational name for a maker or player of bass viols, from Polish, Ukrainian, and Yiddish bas ‘bass viol’.German : see Basse.
Boy/Male
Indian
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Boy/Male
Indian
Vast, Spacious, One who stretches, Enlarges
BASIS THEOREM
BASIS THEOREM
Surname or Lastname
English and Jewish (Ashkenazic)
English and Jewish (Ashkenazic) : patronymic from the personal name Mark.
Boy/Male
French
Of Mars; the god of war.
Girl/Female
English American
derived from Madeline: Woman from Magdala.
Boy/Male
Welsh
Legendary son of Poch.
Boy/Male
Indian, Kannada, Tamil
Warm
Girl/Female
Celtic
A mythical queen.
Male
Japanese
(穂高) Japanese name, possibly HOTAKA means "step by step," derived from the name of the highest peak in what is known as the Japanese Alps.Â
Girl/Female
Sikh
Eternal Lord
Boy/Male
Tamil
Palace, One of the three worlds
Boy/Male
Hindu
Ratined gold
BASIS THEOREM
BASIS THEOREM
BASIS THEOREM
BASIS THEOREM
BASIS THEOREM
n.
The quantity contained in a basin.
n.
The name given to several aromatic herbs of the Mint family, but chiefly to the common or sweet basil (Ocymum basilicum), and the bush basil, or lesser basil (O. minimum), the leaves of which are used in cookery. The name is also given to several kinds of mountain mint (Pycnanthemum).
n.
An isolated or circumscribed formation, particularly where the strata dip inward, on all sides, toward a center; -- especially applied to the coal formations, called coal basins or coal fields.
a.
Hence, basic; metallic; not acid; -- opposed to negative, and said of metals, bases, and basic radicals.
pl.
of Bass
n.
The striped bass. See Bass.
n.
The southern, red, or channel bass (Sciaena ocellata). See Redfish.
n.
Species of Serranus, the sea bass and rock bass. See Sea bass.
n.
A basin.
n.
The striped bass. See Bass.
n.
The deepest pedal stop, or the lowest tones of an organ; the fundamental or ground bass.
n.
The two American fresh-water species of black bass (genus Micropterus). See Black bass.
a.
One who sings, or the instrument which plays, bass.
pl.
of Oasis
pl.
of Basis
a.
A bass, or deep, sound or tone.
v. & a.
Fixed foundation; established basis.