Absolute irreducibility
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example, is absolutely irreducible, but while is irreducible over the integers and the reals, it is reducible over the complex numbers as 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.
Absolutely irreducible is also applied, with the same meaning, to linear representations of algebraic groups.
In all cases, being absolutely irreducible is the same as being irreducible over the algebraic closure 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 irreducible two-dimensional representation of the symmetric group S3 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, 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
- The algebraic variety given by the equation