Absolute irreducibility

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, 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,[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.

  1. ^ Borevich, Z. I.; Shafarevich, I. R. (1986), Number theory, Pure and Applied Mathematics, vol. 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. ^ 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.