Editing Factorization (section)

==Unique factorization domains==

The integers and the polynomials over a [[field (mathematics)|field]] share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a [[unit (ring theory)|unit]], ±1 in the case of integers) and a product of [[irreducible element]]s ([[prime number]]s, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. [[Integral domain]]s which share this property are called [[unique factorization domain]]s (UFD).

[[Greatest common divisor]]s exist in UFDs, but not every integral domain in which greatest common divisors exist (known as a [[GCD domain]]) is a UFD. Every [[principal ideal domain]] is a UFD.

A [[Euclidean domain]] is an integral domain on which is defined a [[Euclidean division]] similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD.

In a Euclidean domain, Euclidean division allows defining a [[Euclidean algorithm]] for computing greatest common divisors. However this does not imply the existence of a factorization algorithm. There is an explicit example of a [[field (mathematics)|field]] {{mvar|F}} such that there cannot exist any factorization algorithm in the Euclidean domain {{math|''F''[''x'']}} of the univariate polynomials over {{mvar|F}}.