Editing Irreducible polynomial (section)

==Algorithms==
{{main|Factorization of polynomials}}
The unique factorization property of polynomials does not mean that the factorization of a given polynomial may always be computed. Even the irreducibility of a polynomial may not always be proved by a computation: there are fields over which no [[algorithm]] can exist for [[decision procedure|deciding]] the irreducibility of arbitrary polynomials.<ref>
{{citation
 |last1=Fröhlich |first1=A.
 |last2=Shepherson |first2=J.C.
 |title = On the factorisation of polynomials in a finite number of steps
 |journal = Mathematische Zeitschrift
 |volume = 62 |issue=1 |year = 1955
 |pages = 331–4
 |issn = 0025-5874
 |doi = 10.1007/BF01180640
 |s2cid = 119955899
}}</ref> 

Algorithms for factoring polynomials and deciding irreducibility are known and implemented in [[computer algebra system]]s for polynomials over the integers, the rational numbers, [[finite field]]s and [[finitely generated field extension]] of these fields. All these algorithms use the algorithms for [[factorization of polynomials over finite fields]].