Absolutely irreducible

In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients. In both 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"₃ of order 6, originally defined over the field of rational numbers, 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

:: $x^2\; +\; y^2\; =\; 1$

: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 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