Editing Irreducible polynomial (section)

==Unique factorization property==
{{main|Unique factorization domain}}

Every polynomial over a field {{math|''F''}} may be factored into a product of a non-zero constant and a finite number of irreducible (over {{math|''F''}}) polynomials. This decomposition is unique [[up to]] the order of the factors and the multiplication of the factors by non-zero constants whose product is 1.

Over a [[unique factorization domain]]  the same theorem is true, but is more accurately formulated by using the notion of primitive polynomial. A [[primitive polynomial (ring theory)|primitive polynomial]] is a polynomial over a unique factorization domain, such that 1 is a [[greatest common divisor]] of its coefficients. 

Let {{math|''F''}} be a unique factorization domain. A non-constant irreducible polynomial over {{math|''F''}} is primitive. A primitive polynomial over {{math|''F''}} is irreducible over {{math|''F''}} if and only if it is irreducible over the [[field of fractions]] of {{math|''F''}}. Every polynomial over {{math|''F''}} may be decomposed into the product of a non-zero constant and a finite number of non-constant irreducible primitive polynomials. The non-zero constant may itself be decomposed into the product of a [[unit (ring theory)|unit]] of {{math|''F''}} and a finite number of [[irreducible element]]s of {{math|''F''}}. Both factorizations are unique up to the order of the factors and the multiplication of the factors by a unit of {{math|''F''}}.

This is this theorem which motivates that the definition of ''irreducible polynomial over a unique factorization domain'' often supposes that the polynomial is non-constant. 

All [[algorithm]]s which are presently [[Implementation#Computer science|implemented]] for factoring polynomials over the [[integer]]s and over the [[rational number]]s use this result (see [[Factorization of polynomials]]).