In mathematics, a

multivariate 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 ...

defined 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 rat ...

s is absolutely irreducible if it is irreducible 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 form ...

.. 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 Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine com ...

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 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 ...

of ''K''. In other words, an absolutely irreducible algebraic set is a synonym of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

,. which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning, to

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

s of

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

s. In all cases, being absolutely irreducible is the same as being irreducible over the

algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...

of the ground field.

Examples

* A univariate polynomial of degree greater than or equal to 2 is never absolutely irreducible, due to 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 ...

. * The irreducible two-dimensional representation of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

''S''₃ of order 6, originally defined over the field of

s, is absolutely irreducible. * The representation of the circle group by rotations in the plane is irreducible (over the field of real numbers), but is not absolutely irreducible. After extending the field to complex numbers, it splits into two irreducible components. This is to be expected, since the circle group is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

and it is known that all irreducible representations of commutative groups over an algebraically closed field are one-dimensional. * The real algebraic variety defined by the equation ::

x^2 + y^2 = 1

:is absolutely irreducible. It is the ordinary

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

over the reals and remains an irreducible

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

over the field of complex numbers. Absolute irreducibility more generally holds over any field not of characteristic two. In characteristic two, the equation is equivalent to (''x'' + ''y'' −1)² = 0. Hence it defines the double line ''x'' + ''y'' =1, which is a non-reduced scheme. * The algebraic variety given by the equation ::

x^2 + y^2 = 0

:is not absolutely irreducible. Indeed, the left hand side can be factored as ::

x^2 + y^2 = (x+yi)(x-yi),

where

i

is a square root of −1. :Therefore, this algebraic variety consists of two lines intersecting at the origin and is not absolutely irreducible. This holds either already over the ground field, if −1 is a square, or over the quadratic extension obtained by adjoining ''i''.

References

{{reflist Algebraic geometry Representation theory