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Polynomial without nontrivial factorization
mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property
Irreducible_polynomial
Sufficient condition for polynomial irreducibility
Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not
Eisenstein's_criterion
Algebraic structure
product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite
Finite_field
computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
About products of primitive polynomials
primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has
Gauss's_lemma_(polynomials)
Computational method
of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors
Factorization_of_polynomials
Polynomial associated with a matrix
characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
Uniform coding for primitive elements of all finite fields
In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define
Conway polynomial (finite fields)
Conway_polynomial_(finite_fields)
Rational fractions as sums of simple terms
p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (i.e. not factorizable into polynomials of positive
Partial fraction decomposition
Partial_fraction_decomposition
Irreducible polynomial whose roots are nth roots of unity
{\displaystyle n} -th cyclotomic polynomial, for any positive integer n {\displaystyle n} , is the unique irreducible polynomial with integer coefficients that
Cyclotomic_polynomial
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Index of articles associated with the same name
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 over
Irreducibility_(mathematics)
Algebraic structure where all polynomials have roots
only irreducible polynomials in the polynomial ring F[x] are those of degree one. The assertion "the polynomials of degree one are irreducible" is trivially
Algebraically_closed_field
Type of algebraic field extension
F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditions for the irreducible polynomial f to be separable:
Separable_extension
Minimal polynomial of a primitive element in a finite field
GF(pm). Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. A primitive polynomial must have a non-zero constant
Primitive polynomial (field theory)
Primitive_polynomial_(field_theory)
Result in number theory, concerning irreducible polynomials
theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number
Hilbert's irreducibility theorem
Hilbert's_irreducibility_theorem
Type of mathematical expression
polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials)
Polynomial
Arithmetic in a field with a finite number of elements
There is at least one irreducible polynomial for which x is a primitive element. In other words, for a primitive polynomial, the powers of x generate
Finite_field_arithmetic
z]}{(y^{2}z-x(x-z)(x-2z))}}\right)} are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial
Hyperconnected_space
Mathematical group
definition of the Galois group comes from the Galois group of an irreducible polynomial f ∈ F [ x ] {\displaystyle f\in F[x]} . If there is a field K /
Galois_group
Error-detecting code for detecting data changes
misconception is that the "best" CRC polynomials are derived from either irreducible polynomials or irreducible polynomials times the factor 1 + x, which adds
Cyclic_redundancy_check
Counts the number of necklaces of n colored beads picked from α available colors
algebra and the number of irreducible polynomials over a finite field. The necklace polynomials are a family of polynomials M n ( k ) {\displaystyle M_{n}(k)}
Necklace_polynomial
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
Factorization algorithm
collapse to an even smaller field, it is sufficient that f is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers
General_number_field_sieve
Characterization of how many integers are prime
Nn be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from
Prime_number_theorem
Cubic equation unsolvable in real radicals
it was shown in the 19th century that, if the polynomial has rational coefficients and is irreducible over the rational numbers, the roots cannot be
Casus_irreducibilis
Theorem on polynomial roots modulo prime powers
nonzero element of ( R / m ) {\displaystyle (R/{\mathfrak {m}})} and irreducible polynomials that are monic (that is, their leading coefficients are 1). Hensel's
Hensel's_lemma
Curve defined as zeros of polynomials
that is considered. If the defining polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise,
Algebraic_curve
Polynomial with 1 as leading coefficient
stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials. Vieta's formulas
Monic_polynomial
Mathematical object studied in the field of algebraic geometry
require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic
Algebraic_variety
Type of algebraic field extension
normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors over L. This
Normal_extension
Manifold or algebraic variety of dimension n in a space of dimension n+1
\ldots ,x_{n})=0,} where p is a multivariate polynomial. Generally the polynomial is supposed to be irreducible. When this is not the case, the hypersurface
Hypersurface
Number with an integer power equal to 1
by definition, the roots of the polynomial xn − 1, and are thus algebraic numbers. As this polynomial is not irreducible (except for n = 1), the primitive
Root_of_unity
Algebraic structure
irreducible polynomial over K {\displaystyle K} has no multiple roots in any field extension L / K {\displaystyle L/K} . Every irreducible polynomial
Perfect_field
Mathematical concept in polynomial theory
resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root
Resultant
Polynomial coprime with its derivative
P(x) is an additive polynomial. Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients
Separable_polynomial
Mathematical term; type of polynomial transformation
{\displaystyle P(t)} a polynomial over K {\displaystyle K} . If P {\displaystyle P} is irreducible, then the quotient ring of the polynomial ring K [ t ] {\displaystyle
Tschirnhaus_transformation
Factorization under function composition
indecomposable polynomials or sometimes prime polynomials (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials).
Polynomial_decomposition
Field theory result
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 a field F
Abel's_irreducibility_theorem
Equation for the real part of a root of unity
/n)\right)=0} . For every n, the polynomial Ψ n ( x ) {\displaystyle \Psi _{n}(x)} is monic, has integer coefficients, and is irreducible over the integers and the
Minimal polynomial of 2cos(2pi/n)
Minimal_polynomial_of_2cos(2pi/n)
determinant, then the resulting multivariable polynomial factors as a product of n irreducible polynomials, where n is the number of conjugacy classes of
Frobenius_determinant_theorem
Transformation of a polynomial induced by a transformation of its roots
computer. If the polynomial P is irreducible, then either the resulting polynomial Q is irreducible, or it is a power of an irreducible polynomial. Let α {\displaystyle
Polynomial_transformation
Pair of polynomial sequences
2n+2}\Psi _{d}(2x).} From the irreducibility of the polynomials Φ n ( x ) {\displaystyle \Phi _{n}(x)} it follows that the polynomials Ψ n ( x ) {\displaystyle
Chebyshev_polynomials
Family of polynomials
Swinnerton-Dyer polynomials are a family of polynomials, introduced by Peter Swinnerton-Dyer, that serve as examples where polynomial factorization algorithms
Swinnerton-Dyer_polynomial
Mathematical invariant of a knot or link
the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional)
Jones_polynomial
linearised polynomial L(x) over Fq is symbolically irreducible if and only if its conventional q-associate l(x) is irreducible over Fq. Every q-polynomial L(x)
Linearised_polynomial
Locus of the zeros of a polynomial of degree two
irreducible polynomial of degree two in D + 1 variables; for example, D=1 is the case of conic sections (plane curves). When the defining polynomial is
Quadric
Commutative ring with no zero divisors other than zero
− x ( x − 1 ) ( x − 2 ) {\displaystyle y^{2}-x(x-1)(x-2)} is an irreducible polynomial. The ring Z [ x ] / ( x 2 − n ) ≅ Z [ n ] {\displaystyle \mathbb
Integral_domain
(Mathematical) decomposition into a product
arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up
Factorization
Mathematical technique
polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p
Matrix factorization of a polynomial
Matrix_factorization_of_a_polynomial
exponential polynomials have been studied in connection with real decision problems and complexity theory. Algorithms for computing irreducible components
Exponential_polynomial
Polynomial whose coefficients are all 1 or −1
Littlewood polynomial is irreducible with probability at least 1 − n − c {\displaystyle 1-n^{-c}} . For reciprocal and skew-reciprocal Littlewood polynomials, Hokken
Littlewood_polynomial
A non-constant polynomial with coefficients in a field is said to be eventually stable if the number of irreducible factors of the n {\displaystyle n}
Eventually_stable_polynomial
Function of the coefficients of a polynomial that gives information on its roots
non-constant polynomial). In nonzero characteristic p, the discriminant is zero if and only if the polynomial is not square-free or it has an irreducible factor
Discriminant
Mathematical construct in computer algebra
(respect. one-step lead reductions) until getting a polynomial that is irreducible (resp. lead-irreducible) by G. It is sometimes called a normal form of f
Gröbner_basis
Natural number
simple 32-stage cycling shift register; also number of binary irreducible polynomials whose degree divides 32 136,279,841 = The largest known Mersenne
100,000,000
Analogue of a prime number in a commutative ring
numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept that is the
Prime_element
Polynomial with no repeated root
In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically
Square-free_polynomial
Type of symmetric polynomials in mathematics
symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations
Schur_polynomial
Standard for the encryption of electronic data
treated as coefficients of polynomial of order x7. Addition is simply XOR. Multiplication is modulo irreducible polynomial x8 + x4 + x3 + x + 1. If processed
Advanced_Encryption_Standard
Equations of degree 5 or higher cannot be solved by radicals
general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give
Abel–Ruffini_theorem
Abstraction of linear independence of vectors
nor the empty set. An irreducible separator is a non-empty separator that contains no other non-empty separator. The irreducible separators partition the
Matroid
Method for computing the relation of two integers with their greatest common divisor
root of an irreducible polynomial of degree d. A simple algebraic extension L of a field K, generated by the root of an irreducible polynomial p of degree
Extended_Euclidean_algorithm
Combinatorial object in representation theory
parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations
Young_tableau
In algebra, element without non-trivial factors
{-5}}\right)=9,} but 3 does not divide either of the two factors. Irreducible polynomial Consider p {\displaystyle p} a prime element of R {\displaystyle
Irreducible_element
Field generated by all rupture-fields of a polynomial over a field
^{2}=2\alpha ^{5}+\alpha ^{2}+2\alpha .} Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is given
Splitting_field
Concept in abstract algebra
/ F {\displaystyle E/F} . It is unique and irreducible over F {\displaystyle F} . If the zero polynomial is the only member of J α {\displaystyle J_{\alpha
Minimal polynomial (field theory)
Minimal_polynomial_(field_theory)
Integral polynomial
theory, a Kazhdan–Lusztig polynomial P y , w ( q ) {\displaystyle P_{y,w}(q)} is a member of a family of integral polynomials introduced by David Kazhdan
Kazhdan–Lusztig_polynomial
Greatest common divisor of polynomials
GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Topology on prime ideals and algebraic varieties
existence of non-linear irreducible polynomials. In this case, the spectrum consists of one closed point for each monic irreducible polynomial, and a generic point
Zariski_topology
In mathematics, invariant of square matrices
generic determinant that is useful in some contexts is that it is an irreducible polynomial (See Determinantal ideal § Properties for a more general property)
Determinant
Polynomial with reversed root positions
{\displaystyle x+1} , hence is not irreducible if its degree is > 1 {\displaystyle >1} . A self-reciprocal polynomial is also called palindromic because
Reciprocal_polynomial
Type of complex number
cos 3π/7, and cos 5π/7 satisfy 8x3 − 4x2 − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic
Algebraic_number
Theorem in algebraic number theory
_{g}(x)^{e_{g}}\mod p,} where π i ( x ) {\displaystyle \pi _{i}(x)} are monic irreducible polynomials in F p [ x ] {\displaystyle \mathbb {F} _{p}[x]} . Then, the ideal
Dedekind–Kummer_theorem
Natural number
prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis
116_(number)
Concept in number theory
certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization
Aurifeuillean_factorization
Type of block code
{\displaystyle (1+x)} . The polynomial ( 1 + x ) {\displaystyle (1+x)} is irreducible in the polynomial ring, and hence the code is an irreducible code. The idempotent
Cyclic_code
Polynomial equation of degree 3
cases where no cube root is needed, that is when the cubic polynomial is not irreducible; this includes the case 4 p 3 + 27 q 2 = 0. {\displaystyle 4p^{3}+27q^{2}=0
Cubic_equation
Field extension generated by a one element
that every element of L is equal to an irreducible fraction of polynomials in θ, and that two such irreducible fractions are equal if and only if one
Simple_extension
Subset (often algebraic set) that is not the union of subsets of the same nature
algebraic set is defined as the set of the zeros of an ideal in a polynomial ring. An irreducible algebraic set, more commonly known as an algebraic variety
Irreducible_component
Theorem about polynomials
worked around by considering only irreducible polynomials; any real polynomial of odd degree must have an irreducible factor of odd degree, which (having
Complex conjugate root theorem
Complex_conjugate_root_theorem
Condition for polynomials to be unfactorable
Perron's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in Z [ x ] {\displaystyle \mathbb {Z} [x]} —that is, for
Perron's irreducibility criterion
Perron's_irreducibility_criterion
mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example
Absolute_irreducibility
lemma (polynomial) Irreducible polynomial Eisenstein's criterion Primitive polynomial Fundamental theorem of algebra Hurwitz polynomial Polynomial transformation
List_of_polynomial_topics
Graph of the frequency response of a control system
{\displaystyle x_{n}} or y n {\displaystyle y_{n}} . In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude
Bode_plot
Polynomial in which all coefficients are one
one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are
All_one_polynomial
Invariant of polynomial roots
the elementary symmetric polynomials. In other words, RG is an irreducible polynomial in Y whose coefficients are polynomial in the coefficients of F
Resolvent_(Galois_theory)
Type of algebraic number
absolute value. For example, the larger of the two roots of the irreducible polynomial x 2 − 3 x + 1 {\displaystyle x^{2}-3x+1} is a Perron number. Perron
Perron_number
Type of hash function
The hash is the remainder after the division of that polynomial by an irreducible polynomial over GF(2). It is possible to update a Rabin fingerprint
Rolling_hash
invented by Conway used to describe polyhedra Conway polynomial (finite fields) – an irreducible polynomial used in finite field theory Conway puzzle – a packing
List of things named after John Horton Conway
List_of_things_named_after_John_Horton_Conway
Generalization of elliptic integrals
{\displaystyle F(x,w)=0,} where F ( x , w ) {\displaystyle F(x,w)} is an irreducible polynomial in w {\displaystyle w} , F ( x , w ) ≡ φ n ( x ) w n + ⋯ + φ 1 (
Abelian_integral
Special-purpose integer factorization algorithm
number fields. Let n be the integer we want to factor. We pick an irreducible polynomial f with integer coefficients, and an integer m such that f(m)≡0 (mod
Special_number_field_sieve
Algebraic structure with addition, multiplication, and division
generated by a single polynomial f in the polynomial ring R = E[X] (over a field E) is maximal if and only if f is irreducible in E, i.e., if f cannot
Field_(mathematics)
Ideal in a ring which has properties similar to prime elements
(p)} . For example, take an irreducible polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} in a polynomial ring F [ x 1 , … , x n ] {\displaystyle
Prime_ideal
Mathematical polynomial factorization
fourth cyclotomic polynomial. As with the cyclotomic polynomials more generally, Φ 4 {\displaystyle \Phi _{4}} is an irreducible polynomial, so this factorization
Sophie_Germain's_identity
Mathematical function
satisfying a polynomial equation P ( x , y ) = 0 {\displaystyle P(x,y)=0} where P ( x , y ) {\displaystyle P(x,y)} is an irreducible polynomial of two variables
Algebraic_function
univariate polynomial ring over R. An irreducible element r in R[X] is either an irreducible element in R or an irreducible primitive polynomial. If r is
Primitive_part_and_content
Special mathematical functions defined on the surface of a sphere
is an irreducible representation of SO(3). The elements of Hℓ arise as the restrictions to the sphere of elements of Aℓ: harmonic polynomials homogeneous
Spherical_harmonics
Fully simplified fraction
numerator and the denominator are coprime polynomials. Every rational number can be represented as an irreducible fraction with positive denominator in exactly
Irreducible_fraction
Conjecture in number theory
at which a given set of polynomials all have prime values. For a set of m {\displaystyle m} distinct irreducible polynomials f 1 , … , f m {\displaystyle
Bateman–Horn_conjecture
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IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
IRREDUCIBLE POLYNOMIAL
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