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Index of articles associated with the same name
In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored
Irreducibility_(mathematics)
Topics referred to by the same term
explanation. Irreducibility may also refer to: Biological irreducibility, a creationist objection to evolution Irreducibility (mathematics), a concept
Irreducibility (disambiguation)
Irreducibility_(disambiguation)
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unpredictable phenomenon in complex systems
Felipe Cucker and Stephen Smale (2007), The Japanese Journal of Mathematics, The Mathematics of Emergence Delsemme, Armand (1998), Our Cosmic Origins: From
Emergence
ISBN 1416594434. Ransford, H. Chris, God and the Mathematics of Infinity: What Irreducible Mathematics Says about Godhood, Columbia University Press, 2017
Mathematics_and_God
Polynomial without nontrivial factorization
Eisenstein's criterion Perron's irreducibility criterion Hilbert's irreducibility theorem Cohn's irreducibility criterion Irreducible component of a topological
Irreducible_polynomial
Fully simplified fraction
a/b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction"
Irreducible_fraction
Subset (often algebraic set) that is not the union of subsets of the same nature
definition of irreducibility and irreducible components extends immediately to schemes. In a Hausdorff space, the irreducible subsets and the irreducible components
Irreducible_component
Awarded every year by the American Mathematical Society
Applied Mathematics (2nd ed.). New York: Interscience Publishers. ISBN 9780471720409. Kostant, Bertram (1975). "On the existence and irreducibility of certain
Leroy_P._Steele_Prize
Sufficient condition for a polynomial to be unfactorable
Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} —that is, for
Cohn's irreducibility criterion
Cohn's_irreducibility_criterion
Sufficient condition for polynomial irreducibility
integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little
Eisenstein's_criterion
Mathematics is a broad subject that is commonly divided in many areas or branches that may be defined by their objects of study, by the used methods,
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Set with associative invertible operation
In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set. The following
Group_(mathematics)
Type of group and algebra representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or
Irreducible_representation
Branch of applied mathematics
broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics. The use of mathematics in the service
Mathematical_economics
Field theory result
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if f(x) is a polynomial over
Abel's_irreducibility_theorem
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For
Absolute_irreducibility
Number of "holes" of a surface
In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface.
Genus_(mathematics)
248-dimensional exceptional simple Lie group
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same
E8_(mathematics)
In mathematics, invariant of square matrices
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the
Determinant
Algebraic structure with addition, multiplication, and division
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on
Field_(mathematics)
In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided R P 2 {\displaystyle \mathbb {R} P^{2}} (real projective
P2-irreducible_manifold
Irreducible fraction
multiplicative inverse of 2. It is an irreducible fraction with a numerator of 1 and a denominator of 2. It often appears in mathematical equations, recipes and measurements
One_half
irreducible, where k ¯ {\displaystyle {\overline {k}}} denotes an algebraic closure of k. X × k k s {\displaystyle X\times _{k}k_{s}} is irreducible for
Geometrically (algebraic geometry)
Geometrically_(algebraic_geometry)
Tool to track locally defined data attached to the open sets of a topological space
Look up sheaf in Wiktionary, the free dictionary. In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian
Sheaf_(mathematics)
Argument by proponents of intelligent design
evolution by mixing and matching genes from various hosts. Arguments for irreducibility often assume that things started out the same way they ended up—as we
Irreducible_complexity
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two
Hyperconnected_space
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
52-dimensional exceptional simple Lie group
In mathematics, F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The
F4_(mathematics)
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals
Irreducible_ideal
Limiting case which is different from the rest of the class
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than)
Degeneracy_(mathematics)
Number of times an object must be counted for making true a general formula
Look up multiplicity in Wiktionary, the free dictionary. In mathematics, the multiplicity of a member of a multiset is the number of times it appears
Multiplicity_(mathematics)
Generalisation of a sheaf; a fibered category that admits effective descent
MR 0399094, S2CID 122887093 Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de
Stack_(mathematics)
Counterintuitive mathematical object
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand
Pathological_(mathematics)
Unique knot with a crossing number of four
joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the figure-eight knot is
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
In algebra, element without non-trivial factors
In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product
Irreducible_element
Number divisible only by 1 and itself
Pages from year three of a mathematical blog. Graduate Studies in Mathematics. Vol. 117. Providence, RI: American Mathematical Society. pp. 82–86. doi:10
Prime_number
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Topics referred to by the same term
reduction: see smelting Reducible as the opposite of irreducible (mathematics) Reduction (mathematics), the rewriting of an expression into a simpler form
Reduction
Mathematical object studied in the field of algebraic geometry
varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra
Algebraic_variety
Multi-lobed plane curve
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates
Rose_(mathematics)
Group that admits a formal description in terms of reflections
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic
Coxeter_group
Mathematical structure
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes
Building_(mathematics)
Analogue of a prime number in a commutative ring
either b or c is a unit, while several non-equivalent definitions of irreducibility of varying strength exist for elements of general commutative rings
Prime_element
In the branch of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored
Subdirectly irreducible algebra
Subdirectly_irreducible_algebra
Generalization of vector spaces from fields to rings
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative)
Module_(mathematics)
Norwegian mathematician (1802–1829)
than Niels; however, a new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818. He gave the students mathematical tasks to do at home. He saw
Niels_Henrik_Abel
Area of mathematics
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
notions of topological irreducibility and algebraic irreducibility of representations of C*-algebras. It implies that, for irreducible representations of
Kadison_transitivity_theorem
Geometric arrangements of points, foundational to Lie theory
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental
Root_system
133-dimensional exceptional simple Lie group
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133;
E7_(mathematics)
Set whose pairs have minima and maxima
completely join irreducible (or ∨ {\displaystyle \vee } -irreducible). The dual notion is meet irreducibility ( ∧ {\displaystyle \wedge } -irreducible). For example
Lattice_(order)
78-dimensional exceptional simple Lie group
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}}
E6_(mathematics)
Relation of degree three
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations
Ternary_relation
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a
List of irreducible Tits indices
List_of_irreducible_Tits_indices
U.S. naval officer and computer scientist (1906–1992)
types of irreducibility criteria", Bull. Amer. Math. Soc. 40 (1934) 216 "New types of irreducibility criteria". Bulletin of the American Mathematical Society
Grace_Hopper
Hungarian and American mathematician and physicist (1903–1957)
solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative
John_von_Neumann
Topics referred to by the same term
reductive group is finitely generated Hilbert's irreducibility theorem, in number theory, concerning irreducible polynomials Hilbert's Nullstellensatz, the
Hilbert's_theorem
Mathematical structures that allow quantum mechanics to be explained
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
1995 publication in mathematics
the satisfaction of the mathematical community. The corrected proof was published in 1995 in the journal Annals of Mathematics in the form of two articles
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Mathematical idealization of the surface of a body
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be
Surface_(mathematics)
Number of ways to pair up n objects
telephone numbers give the sum of the degrees of the irreducible representations. In the mathematics of chess, the telephone numbers count the number of
Telephone number (mathematics)
Telephone_number_(mathematics)
Group of gauge symmetries in Yang–Mills theory
configuration space of quantum gauge theory. Gauge symmetry (mathematics) Gauge theory Gauge theory (mathematics) Principal bundle Mitter, P., Viallet, C., On the
Gauge_group_(mathematics)
Random process independent of past history
there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov
Markov_chain
English mathematician and philosopher (1861–1947)
economics, and psychology. In his early career Whitehead wrote primarily on mathematics, logic, and physics. He wrote the three-volume Principia Mathematica
Alfred_North_Whitehead
Three-holed sphere
In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks
Pair_of_pants_(mathematics)
Concept in mathematics
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic
Fully irreducible automorphism
Fully_irreducible_automorphism
Russian mathematician
approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020 (with Hillel Furstenberg), becoming
Grigory_Margulis
Topics referred to by the same term
rest of the protein chain Social domain, a concept in sociology Domain (mathematical analysis), an open connected set Domain (ring theory), a non-trivial
Domain
Simple Lie group; the automorphism group of the octonions
In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak
G2_(mathematics)
Skeletonized version of algebraic geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication
Tropical_geometry
Topological space whose topology is fully captured by its lattice of open sets
In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of
Sober_space
Natural number
Groups". Contributions to Discrete Mathematics. 5 (2). Alberta, CA: University of Calgary Department of Mathematics and Statistics: 27. doi:10.11575/cdm
27_(number)
In mathematics, and especially the discipline of representation theory, the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes
Frobenius–Schur_indicator
Manifold or algebraic variety of dimension n in a space of dimension n+1
Mathematical Society 22(1): 301,2 Lima, Elon L. (1988). "The Jordan-Brouwer separation theorem for smooth hypersurfaces". The American Mathematical Monthly
Hypersurface
German mathematician (1880–1975)
"Oskar Perron", MacTutor History of Mathematics Archive, University of St Andrews Oskar Perron at the Mathematics Genealogy Project Gabriele Dörflinger:
Oskar_Perron
Argentine-American mathematician
Mathematics (autobiographical essay 2021) (online) Infinity, Incompleteness, Irreducibility (course notes 2024) (online) Gregory Chaitin (2007), Algorithmic information
Gregory_Chaitin
American Jewish mathematician
Journal of Mathematics. 85 (3): 327–404. doi:10.2307/2373130. JSTOR 2373130. Kostant, Bertram (1969). "On the existence and irreducibility of certain
Bertram_Kostant
Algebraic structure
factors) into a product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials
Finite_field
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)
theorem (number theory) Chowla–Mordell theorem (number theory) Cohn's irreducibility criterion (polynomials) Critical line theorem (number theory) Davenport–Schmidt
List_of_theorems
American scientist (1839–1914)
of Mathematics Peirce wrote drafts for an introductory textbook, with the working title The New Elements of Mathematics, that presented mathematics from
Charles_Sanders_Peirce
Type of transfinite numbers
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map
Epsilon_number
German mathematician (1862–1943)
1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of all time. Hilbert discovered
David_Hilbert
Harmonic functions as solutions to Laplace's equation
In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" dates from 19th-century physics
Potential_theory
event. This reformulates in more geometric terms the classical Hilbert irreducibility theorem. A thin set, in general, is a subset of a finite union of thin
Thin_set_(Serre)
Mathematical function generalizing the determinant and permanent
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant
Immanant
Concerns the decomposition of representations of a finite group into irreducible pieces
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations
Maschke's_theorem
Book by Stephen Wolfram
science. For instance, he argues that the concept of computational irreducibility (that some complex computations are not amenable to short-cuts and cannot
A_New_Kind_of_Science
Group representation
In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ* is defined over the dual
Dual_representation
Equivalence of distributive lattices and set families
similarly named results, see Birkhoff's theorem (disambiguation). In mathematics, Birkhoff's representation theorem for distributive lattices states that
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Open problem on 3x+1 and x/2 functions
Unsolved problem in mathematics For even numbers, divide by 2; For odd numbers, multiply by 3 and add 1. With enough repetition, do all positive integers
Collatz_conjecture
Ratio of two numbers
negative produces a positive, −1/−2 represents positive one-half. In mathematics a rational number is a number that can be represented by a fraction of
Fraction
Unsolved problem in mathematics
functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group
Inverse_Galois_problem
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. A (meet-)irreducible ring is a ring
Irreducible_ring
Cubic equation unsolvable in real radicals
Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved
Casus_irreducibilis
Type of mathematical expression
finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials)
Polynomial
Calculation of complex statistical distributions
point-to-point transitions have zero probability. In this case, φ-irreducibility generalizes irreducibility by using a reference measure φ on the measurable space
Markov_chain_Monte_Carlo
Matrices named after Élie Cartan
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices
Cartan_matrix
in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers Zariski, Oscar (1948), "Analytical irreducibility of normal varieties"
Analytically_irreducible_ring
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IRREDUCIBILITY MATHEMATICS
IRREDUCIBILITY MATHEMATICS
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