Absolutely irreducible

In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.[1][2][3] For example, x^2+y^2-1 is absolutely irreducible, but x^2+y^2 is irreducible over the integers and the reals, but, as x^2+y^2 = (x+iy)(x-iy), is 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,[4] 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,[5] 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

 x^2 + y^2 = 1
is absolutely irreducible.[3] 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.
 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

  1. Borevich, Z. I.; Shafarevich, I. R. (1986), Number theory, Pure and Applied Mathematics 20, Academic Press, p. 10, ISBN 9780080873329.
  2. Grabmeier, Johannes; Kaltofen, Erich; Weispfenning, Volker (2003), Computer Algebra Handbook: Foundations, Applications, Systems, Springer, p. 26, ISBN 9783540654667.
  3. 1 2 Tucker, Allen B. (2004), Computer Science Handbook (2nd ed.), CRC Press, pp. 8–17 – 8–18, ISBN 9780203494455.
  4. Stepanov, Serguei A. (1994), Arithmetic of Algebraic Curves, Monographs in Contemporary Mathematics, Springer, p. 53, ISBN 9780306110368.
  5. Niederreiter, Harald; Xing, Chaoping (2009), Algebraic Geometry in Coding Theory and Cryptography, Princeton University Press, p. 47, ISBN 9781400831302.
This article is issued from Wikipedia - version of the Tuesday, October 20, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.