View source for Rokhlin's theorem

In [[mathematics]], a [[multivariate polynomial]] defined over the [[rational number]]s is '''absolutely irreducible''' if it is  [[irreducible polynomial|irreducible]] over the [[complex field]]. For example <math>x^2+y^2-1</math> is absolutely irreducible, but <math>x^2+y^2</math> is irreducible over the integers and the reals, but, as <math>x^2+y^2 = (x+iy)(x-iy),</math>  is not absolutely irreducible.

More generally, 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, '''absolutely irreducible algebraic set''' is a synonymous of [[algebraic variety]], 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]]s of [[algebraic group]]s.

In all cases, being absolutely irreducible is the same as being irreducible over the [[algebraic closure]] of the ground field.

== Examples ==
* The irreducible two-dimensional representation of the [[symmetric group]] ''S''<sub>3</sub> of order 6, originally defined over the field of [[rational number]]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]] 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 

:: <math> x^2 + y^2 = 1 </math>

:is absolutely irreducible. It is the ordinary [[circle]] over the reals and remains an irreducible [[conic section]] over the field of complex numbers. Absolute irreducibility more generally holds over any field not of [[characteristic (field)|characteristic]] two. In characteristic two, the equation is equivalent to (''x''&nbsp;+&nbsp;''y''&nbsp;−1)<sup>2</sup> = 0. Hence it defines the double line ''x''&nbsp;+&nbsp;''y''&nbsp;=1, which is a [[reduced ring|non-reduced]] [[scheme (algebraic geometry)|scheme]]. 

* The algebraic variety given by the equation

:: <math> x^2 + y^2 = 0 </math>

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

:: <math> x^2 + y^2 = (x+yi)(x-yi), </math> where <math>i</math> 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==
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[[Category:Algebraic geometry]]
[[Category:Representation theory]]