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In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x² − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, 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 ${\displaystyle \left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)}$ if it is considered as a polynomial with real coefficients. One says that the polynomial x² − 2 is irreducible over the integers but not over the reals.

Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a polynomial over an integral domain R is said to be irreducible if it is not the product of two polynomials that have their coefficients in R, and that are not unit in R. Equivalently, for this definition, an irreducible polynomial is an irreducible element in the rings of polynomials over R. If R 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 of the integral domain. For example, the polynomial ${\displaystyle 2(x^{2}-2)\in \mathbb {Z} }$ is irreducible for the second definition, and not for the first one. On the other hand, ${\displaystyle x^{2}-2}$ is irreducible in ${\displaystyle \mathbb {Z} }$ for the two definitions, while it is reducible in ${\displaystyle \mathbb {R} .}$

A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible. By the fundamental theorem of algebra, a univariate polynomial 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 ${\displaystyle x^{2}+y^{n}-1,}$ for any positive integer n.

A polynomial that is not irreducible is sometimes said to be a reducible polynomial.^[1][2]

Irreducible polynomials appear naturally in the study of polynomial factorization and algebraic field extensions.

It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. 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, an irreducible polynomial is also called a prime polynomial, because it generates a prime ideal.

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 R, is sometimes said to be irreducible (or irreducible over R) if it is an irreducible element of the polynomial ring, 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 of R (the field of rational numbers, if R is the integers). This second definition is not used in this article. The equivalence of the two definitions depends on R.

Simple examples

The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

${\displaystyle {\begin{aligned}p_{1}(x)&=x^{2}+4x+4\,={(x+2)^{2}}\\p_{2}(x)&=x^{2}-4\,={(x-2)(x+2)}\\p_{3}(x)&=9x^{2}-3\,=3\left(3x^{2}-1\right)\,=3\left(x{\sqrt {3}}-1\right)\left(x{\sqrt {3}}+1\right)\\p_{4}(x)&=x^{2}-{\frac {4}{9}}\,=\left(x-{\frac {2}{3}}\right)\left(x+{\frac {2}{3}}\right)\\p_{5}(x)&=x^{2}-2\,=\left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)\\p_{6}(x)&=x^{2}+1\,={(x-i)(x+i)}\end{aligned}}}$

Over the integers, 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 numbers, 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, and, therefore, does not count as a factor).

Over the real numbers, the first five polynomials are reducible, but ${\displaystyle p_{6}(x)}$ is irreducible.

Over the complex numbers, all six polynomials are reducible.

Over the complex numbers

Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducible if and only if its degree is one. This fact is known as the fundamental theorem of algebra 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

${\displaystyle a\left(x-z_{1}\right)\cdots \left(x-z_{n}\right)}$

where ${\displaystyle n}$ is the degree, ${\displaystyle a}$ is the leading coefficient and ${\displaystyle z_{1},\dots ,z_{n}}$ are the zeros of the polynomial (not necessarily distinct, and not necessarily having explicit algebraic expressions).

There are irreducible multivariate polynomials of every degree over the complex numbers. For example, the polynomial

${\displaystyle x^{n}+y^{n}-1,}$

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 polynomials ${\displaystyle ax^{2}+bx+c}$ that have a negative discriminant ${\displaystyle b^{2}-4ac.}$ It follows that every non-constant univariate polynomial can be factored as a product of polynomials of degree at most two. For example, ${\displaystyle x^{4}+1}$ factors over the real numbers as ${\displaystyle \left(x^{2}+{\sqrt {2}}x+1\right)\left(x^{2}-{\sqrt {2}}x+1\right),}$ and it cannot be factored further, as both factors have a negative discriminant: Zdroj:https://en.wikipedia.org?pojem=Reducible_polynomial
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