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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an irreducible polynomial is, roughly speaking, a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s that are accepted for the possible factors, that is, the field to which the
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients, but, as every integer is also a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, and there are two common definitions. Most often, a polynomial over an integral domain is said to be ''irreducible'' if it is not the product of two polynomials that have their coefficients in , and are not
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
in . Equivalently, for this definition, an irreducible polynomial is an irreducible element in the rings of polynomials over . If is a field, the two definitions of irreducibility are equivalent. For the second definition, a polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that both have a positive degree. Equivalently, a polynomial is irreducible if it is irreducible over the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the integral domain. For example, the polynomial 2(x^2-2)\in \Z /math> is irreducible for the second definition, and not for the first one. On the other hand, x^2-2 is irreducible in \Z /math> for the two definitions, while it is reducible in \R A polynomial that is irreducible over any field containing the coefficients is
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the intege ...
. By the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, a
univariate polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
is absolutely irreducible if and only if its degree is one. On the other hand, with several indeterminates, there are absolutely irreducible polynomials of any degree, such as x^2 + y^n - 1, for any positive integer . A polynomial that is not irreducible is sometimes said to be a reducible polynomial. Irreducible polynomials appear naturally in the study of
polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same dom ...
and algebraic field extensions. It is helpful to compare irreducible polynomials to
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. They exhibit many of the general properties of the concept of "irreducibility" that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
, an irreducible polynomial is also called a prime polynomial, because it generates a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
.


Definition

If ''F'' is a field, a non-constant polynomial is irreducible over ''F'' if its coefficients belong to ''F'' and it cannot be factored into the product of two non-constant polynomials with coefficients in ''F''. A polynomial with integer coefficients, or, more generally, with coefficients in a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
''R'', is sometimes said to be ''irreducible'' (or ''irreducible over R'') if it is an irreducible element of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
, that is, it is not invertible, not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in ''R''. This definition generalizes the definition given for the case of coefficients in a field, because, over a field, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero. Another definition is frequently used, saying that a polynomial is ''irreducible over R'' if it is irreducible over the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''R'' (the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, if ''R'' is the integers). This second definition is not used in this article.


Nature of a factor

The absence of an explicit
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
for a factor does not by itself imply that a polynomial is irreducible. When a polynomial is reducible into factors, these factors may be explicit algebraic expressions or implicit expressions. For example, x^2 + 2 can be factored explicitly over the complex numbers as \left(x - \sqrti\right)\left(x + \sqrti\right); however, the Abel–Ruffini theorem states that there are polynomials of any degree greater than 4 for which complex factors exist that have no explicit algebraic expression. Such a factor can be written simply as, say, \left(x - x_1\right), where x_1 is defined implicitly as a particular solution of the equation that sets the polynomial equal to 0. Further, factors of either type can also be expressed as numerical approximations obtainable by
root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
s, for example as \left(x - 1.2837\ldots\right).


Simple examples

The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials: :\begin p_1(x) &= x^2 + 4x + 4\, = \\ p_2(x) &= x^2 - 4\, = \\ p_3(x) &= 9x^2 - 3\, = 3\left(3x^2 - 1\right)\, = 3\left(x\sqrt - 1\right)\left(x\sqrt + 1\right)\\ p_4(x) &= x^2 - \frac\, = \left(x - \frac\right)\left(x + \frac\right)\\ p_5(x) &= x^2 - 2\, = \left(x - \sqrt\right)\left(x + \sqrt\right)\\ p_6(x) &= x^2 + 1\, = \end Over the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers); the last two are irreducible. (The fourth, of course, is not a polynomial over the integers.) Over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, the first two and the fourth polynomials are reducible, but the other three polynomials are irreducible (as a polynomial over the rationals, 3 is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
, and, therefore, does not count as a factor). Over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, the first five polynomials are reducible, but p_6(x) is irreducible. Over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, all six polynomials are reducible.


Over the complex numbers

Over the
complex field In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
, and, more generally, over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, a
univariate polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
is irreducible if and only if its degree is one. This fact is known as the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
in the case of the complex numbers and, in general, as the condition of being algebraically closed. It follows that every nonconstant univariate polynomial can be factored as :a\left(x - z_1\right) \cdots \left(x - z_n\right) where n is the degree, a is the leading coefficient and z_1, \dots, z_n are the zeros of the polynomial (not necessarily distinct, and not necessarily having explicit
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...
s). There are irreducible multivariate polynomials of every degree over the complex numbers. For example, the polynomial :x^n + y^n - 1, which defines a Fermat curve, is irreducible for every positive ''n''.


Over the reals

Over the field of reals, the degree of an irreducible univariate polynomial is either one or two. More precisely, the irreducible polynomials are the polynomials of degree one and the
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
s ax^2 + bx + c that have a negative
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
b^2 - 4ac. It follows that every non-constant univariate polynomial can be factored as a product of polynomials of degree at most two. For example, x^4 + 1 factors over the real numbers as \left(x^2 + \sqrtx + 1\right)\left(x^2 - \sqrtx + 1\right), and it cannot be factored further, as both factors have a negative discriminant: \left(\pm\sqrt\right)^2 - 4 = -2 < 0.


Unique factorization property

Every polynomial over a field may be factored into a product of a non-zero constant and a finite number of irreducible (over ) polynomials. This decomposition is unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
the order of the factors and the multiplication of the factors by non-zero constants whose product is 1. Over a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
the same theorem is true, but is more accurately formulated by using the notion of primitive polynomial. A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of its coefficients. Let be a unique factorization domain. A non-constant irreducible polynomial over is primitive. A primitive polynomial over is irreducible over if and only if it is irreducible over the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of . Every polynomial over may be decomposed into the product of a non-zero constant and a finite number of non-constant irreducible primitive polynomials. The non-zero constant may itself be decomposed into the product of a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
of and a finite number of irreducible elements of . Both factorizations are unique up to the order of the factors and the multiplication of the factors by a unit of . This is this theorem which motivates that the definition of ''irreducible polynomial over a unique factorization domain'' often supposes that the polynomial is non-constant. All
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s which are presently implemented for factoring polynomials over the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s use this result (see
Factorization of polynomials In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same do ...
).


Over the integers and finite fields

The irreducibility of a polynomial over the integers \mathbb is related to that over the field \mathbb_p of p elements (for a prime p). In particular, if a univariate polynomial over \mathbb Z is irreducible over \mathbb_p for some prime p that does not divide the leading coefficient of (the coefficient of the highest power of the variable), then is irreducible over \mathbb (that is, it is not the product of two non-constant polynomials with integer coefficients).
Eisenstein's criterion In mathematics, 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 be factorizable into the product of non-constant polynomials with ...
is a variant of this property where irreducibility over p^2 is also involved. The converse, however, is not true: there are polynomials of arbitrarily large degree that are irreducible over the integers and reducible over every finite field. A simple example of such a polynomial is x^4 + 1. The relationship between irreducibility over the integers and irreducibility modulo ''p'' is deeper than the previous result: to date, all implemented algorithms for factorization and irreducibility over the integers and over the rational numbers use the factorization over finite fields as a
subroutine In computer programming, a function or subroutine is a sequence of program instructions that performs a specific task, packaged as a unit. This unit can then be used in programs wherever that particular task should be performed. Functions may ...
. The number of degree irreducible
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\ ...
s over a field \mathbb_q for a prime power is given by :N(q, n) = \frac\sum_ \mu(d)q^\frac, where is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
. For , such polynomials are commonly used to generate pseudorandom binary sequences. In some sense, almost all polynomials with coefficients zero or one are irreducible over the integers. More precisely, if a version of the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
for Dedekind zeta functions is assumed, the probability of being irreducible over the integers for a polynomial with
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...
coefficients in tends to one when the degree increases.


Algorithms

The unique factorization property of polynomials does not mean that the factorization of a given polynomial may always be computed. Even the irreducibility of a polynomial may not always be proved by a computation: there are fields over which no
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
can exist for deciding the irreducibility of arbitrary polynomials. Algorithms for factoring polynomials and deciding irreducibility are known and implemented in
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
s for polynomials over the integers, the rational numbers,
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s and
finitely generated field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of these fields. All these algorithms use the algorithms for
factorization of polynomials over finite fields In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, bu ...
.


Field extension

The notions of irreducible polynomial and of algebraic field extension are strongly related, in the following way. Let ''x'' be an element of an
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...
''L'' of a field ''K''. This element is said to be ''algebraic'' if it is a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of a nonzero polynomial with coefficients in ''K''. Among the polynomials of which ''x'' is a root, there is exactly one which is monic and of minimal degree, called the minimal polynomial of ''x''. The minimal polynomial of an algebraic element ''x'' of ''L'' is irreducible, and is the unique monic irreducible polynomial of which ''x'' is a root. The minimal polynomial of ''x'' divides every polynomial which has ''x'' as a root (this is Abel's irreducibility theorem). Conversely, if P(X) \in K /math> is a univariate polynomial over a field ''K'', let L = K P(X) be the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of the polynomial ring K /math> by the ideal generated by . Then is a field if and only if is irreducible over . In this case, if is the image of in , the minimal polynomial of is the quotient of by its leading coefficient. An example of the above is the standard definition of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s as \mathbb = \mathbb ;/\left(X^2 + 1\right). If a polynomial has an irreducible factor over , which has a degree greater than one, one may apply to the preceding construction of an algebraic extension, to get an extension in which has at least one more root than in . Iterating this construction, one gets eventually a field over which factors into linear factors. This field, unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a field isomorphism, is called the
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
of .


Over an integral domain

If ''R'' is an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, an element ''f'' of ''R'' that is neither zero nor a unit is called irreducible if there are no non-units ''g'' and ''h'' with ''f'' = ''gh''. One can show that every
prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...
is irreducible;Consider ''p'' a prime that is reducible: ''p'' = ''ab''. Then ''p'' , ''ab'' ⇒ ''p'' , ''a'' or ''p'' , ''b''. Say ''p'' , ''a'' ⇒ ''a'' = ''pc'', then we have: ''p'' = ''ab'' = ''pcb'' ⇒ ''p''(1 − ''cb'') = 0. Because ''R'' is a domain, we have ''cb'' = 1. So ''b'' is a unit, and ''p'' is irreducible. the converse is not true in general but holds in
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
s. The
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''F'' 'x''over a field ''F'' (or any unique-factorization domain) is again a unique factorization domain. Inductively, this means that the polynomial ring in ''n'' indeterminates (over a ring ''R'') is a unique factorization domain if the same is true for ''R''.


See also

*
Gauss's lemma (polynomial) In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statementThis theorem is called a lemma for historical reasons. about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that ...
*
Rational root theorem In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation :a_nx^n+a_x^+\cdots+a_0 = 0 with integer coefficients a_i\in ...
, a method of finding whether a polynomial has a linear factor with rational coefficients *
Eisenstein's criterion In mathematics, 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 be factorizable into the product of non-constant polynomials with ...
* Perron's irreducibility criterion * Hilbert's irreducibility theorem * Cohn's irreducibility criterion *
Irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal ( ...
of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
*
Factorization of polynomials over finite fields In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, bu ...
* * * Casus irreducibilis, the irreducible cubic with three real roots *


Notes


References

* . This classical book covers most of the content of this article. * *
pp. 91
* *
pp. 154


External links

* *

The (Combinatorial) Object Server. {{Polynomials Polynomials Abstract algebra Algebra