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⭐ Core Definition: Irreducible polynomial

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 $x - 2$ 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 $\left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)$ if it is considered as a polynomial with real coefficients. One says that the polynomial $x - 2$ 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 a ring 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 $2(x^{2}-2)\in \mathbb {Z} [x]$ is irreducible for the second definition, and not for the first one. On the other hand, $x^{2}-2$ is irreducible in $\mathbb {Z} [x]$ for the two definitions, while it is reducible in $\mathbb {R} [x].$

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👉 Irreducible polynomial in the context of Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation $h (x, y, t) = 0$ can be restricted to the affine algebraic plane curve of equation $h (x, y, 1) = 0$ . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

If the defining polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its components, that are defined by the irreducible factors.

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⭐ Core Definition: Irreducible polynomial
👉 Irreducible polynomial in the context of Algebraic curve
Irreducible polynomial in the context of Quadric surface
Irreducible polynomial in the context of Absolutely irreducible
Irreducible polynomial in the context of Polynomial factorization

Irreducible polynomial in the context of Quadric surface

In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.

More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension D + 1) is defined as the zero set of an 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 not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

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Irreducible polynomial in the context of 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 integers and the reals, it is reducible over the complex numbers as $x^{2}+y^{2}=(x+iy)(x-iy),$ and thus not absolutely irreducible.

More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K, and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety, which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field.

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Irreducible polynomial in the context 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 domain. Polynomial factorization is one of the fundamental components of computer algebra systems.

The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems:

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Irreducible polynomial in the context of "Algebraic curve"

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⭐ Core Definition: Irreducible polynomial

👉 Irreducible polynomial in the context of Algebraic curve

Irreducible polynomial in the context of Quadric surface

Irreducible polynomial in the context of Absolutely irreducible

Irreducible polynomial in the context of Polynomial factorization