Factorization of polynomials: Difference between revisions

Revision as of 18:43, 18 February 2006

Revision as of 13:42, 24 February 2006

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'''Polynomial factorization''' typically refers to [[factor]]ing a [[polynomial]] into [[irreducible polynomial]]s over a given [[field (mathematics)|field]]. Other factorizations, such as [[square-free polynomial|square-free]] factorization exist, but the irreducible factorization, the most common, is the subject of this article. The factorization depends strongly on the choice of field. For example, the [[fundamental theorem of algebra]], which states that all polynomials with [[complex number|complex]] coefficients have complex roots, implies that a polnomial with [[integer]] [[coefficient]]s can be completely reduced to [[linear function|linear factor]]s over the complex field '''C'''. On the other hand, such a polynomial can may only be reducable to linear and [[quadratic function|quadratic]] factors over the [[real number|real]] field '''R'''. Over the [[rational number]] field '''Q''', it is possible that no factorization at all may be possible. From a more practical vantage point, the fundamental theorem is only an existanceexistence proof that offers little insight into the common problem of actually finding the roots of a given polynomial.

===Factoring over '''Q''' and '''Z'''===

It can be shown that factoring over '''Q''' (the rational numbers) can be reduced to factoring over '''Z''' (the integers). This is a specific example of a more general case — factoring over a [[quotient field]] can be reduced to factoring over the corresponding [[integral ___domain]].

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===Factoring over '''<math>Z_n</math>'''===

Berklekamp's algorithm.

===Factoring over algebraic extensions===

===Bibliography===

* Van der Waerden, ''Algebra'' (1970), trans. Blum and Schulenberger, Frederick Ungar.

[[Category: Polynomials]]