View source for Cohn's irreducibility criterion

{{short description|Sufficient condition for a polynomial to be unfactorable}}
'''Cohn's irreducibility criterion''' is a sufficient condition for a [[polynomial]] to be [[irreducible polynomial|irreducible]] in [[polynomial ring|<math>\mathbb{Z}[x]</math>]]—that is, for it to be unfactorable into the product of lower-[[degree of a polynomial|degree]] polynomials with [[integer]] [[coefficient]]s.

==Statement==
The criterion is often stated as follows:
:If a [[prime number]] <math>p</math> is expressed in [[base 10]] as <math>p = a_m 10^m + a_{m-1} 10^{m-1} +\cdots+ a_1 10 + a_0</math> (where <math>0\leq a_i\leq 9</math>) then the polynomial
::<math>f(x)=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0</math>
:is irreducible in <math>\mathbb{Z}[x]</math>.<ref name=murty/>

The theorem can be generalized to other bases as follows:
:Assume that <math>b \ge 2</math> is a [[natural number]] and <math>p(x) = a_k x^k + a_{k-1} x^{k-1} +\cdots+ a_1 x + a_0</math> is a polynomial such that <math>0\leq a_i \leq b-1</math>. If <math>p(b)</math> is a prime number then <math>p(x)</math> is irreducible in <math>\mathbb{Z}[x]</math>.<ref name=murty/>

==History and extensions==
The base 10 version of the theorem is attributed to Cohn by [[George Pólya|Pólya]] and [[Gábor Szegő|Szegő]] in ''[[Problems and Theorems in Analysis]]''<ref name=Polya>{{cite book |last1=Pólya |first1=George |last2=Szegő  |first2=Gábor |title=Aufgaben und Lehrsätze aus der Analysis, Bd 2 |year=1925 |publisher=Springer, Berlin |oclc=73165700 }} English translation in: {{cite book |last1=Pólya |first1=George |last2=Szegő  |first2=Gábor |title=Problems and theorems in analysis, volume 2 |publisher=Springer |year=2004 |volume=2 |isbn=978-3-540-63686-1 |page=137}}</ref> while the generalization to any base ''b'' is due to Brillhart, [[Michael Filaseta|Filaseta]], and [[Andrew Odlyzko|Odlyzko]].<ref name=Brillhart>{{cite journal |last1=Brillhart |first1=John |authorlink1=John Brillhart |last2=Filaseta |first2=Michael |last3=Odlyzko |first3=Andrew |authorlink3=Andrew Odlyzko |title=On an irreducibility theorem of A. Cohn | journal = Canadian Journal of Mathematics | year = 1981 | volume = 33 | issue = 5 | pages = 1055–1059 | doi = 10.4153/CJM-1981-080-0 |doi-access=free }}</ref> It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn (1894–1940), a student of [[Issai Schur]] who was awarded his doctorate from [[Humboldt University of Berlin|Frederick William University]] in 1921.<ref>[http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=17963 Arthur Cohn's entry at the Mathematics Genealogy Project]</ref><ref>{{cite book|last1=Siegmund-Schultze|first1=Reinhard|title=Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact|date=2009|publisher=Princeton University Press|location=Princeton, N.J.|isbn=9781400831401|page=346}}<!--|accessdate=9 March 2015--></ref>

A further generalization of the theorem allowing coefficients larger than digits was given by Filaseta and Gross.<ref name=Filaseta>{{cite journal |last1=Filaseta |first1=Michael |last2=Gross |first2=Samuel S. |title=49598666989151226098104244512918 | journal = Journal of Number Theory | year = 2014 | volume = 137  | pages = 16–49 | doi = 10.1016/j.jnt.2013.11.001 |doi-access=free }}</ref> In particular, let <math>f(x) </math> be a polynomial with non-negative integer coefficients such that <math>f(10)</math> is prime. If all coefficients are <math> \leq </math> 49598666989151226098104244512918, then <math>f(x)</math> is irreducible over <math>\mathbb{Z}[x]</math>. Moreover, they proved that this bound is also sharp. In other words, coefficients larger than 49598666989151226098104244512918 do not guarantee irreducibility. The method of Filaseta and Gross was also generalized to provide similar sharp bounds for some other bases by Cole, Dunn, and Filaseta.<ref name=Cole>{{cite journal |last1=Cole |first1=Morgan |last2=Dunn |first2=Scott |last3=Filaseta |first3=Michael |title=Further irreducibility criteria for polynomials with non-negative coefficients | journal = Acta Arithmetica | year = 2016 | volume = 175 | pages = 137–181 | doi = 10.4064/aa8376-5-2016 }}</ref>

An analogue of the theorem also holds for [[algebraic function field]]s over [[finite field]]s.<ref name=murty/>

==Converse==
The [[converse (logic)|converse]] of this criterion is that, if ''p'' is an irreducible polynomial with integer coefficients that have [[greatest common divisor]] 1, then there exists a base such that the coefficients of ''p'' form the representation of a prime number in that base. This is the [[Bunyakovsky conjecture]] and its truth or falsity remains an open question.<ref name=murty>{{cite journal | last = Murty | first = Ram | title = Prime Numbers and Irreducible Polynomials | journal = [[American Mathematical Monthly]] | year = 2002 | volume = 109 | issue = 5 | pages = 452–458 | url = http://www.mast.queensu.ca/~murty/murty.pdf | doi = 10.2307/2695645 | jstor = 2695645 | citeseerx = 10.1.1.225.8606 }} (dvi file)</ref>

==See also==

*[[Eisenstein's criterion]]
*[[Perron's irreducibility criterion]]

==References==
<references/>

==External links==
*{{planetmath reference|urlname=ACohnsIrreducibilityCriterion|title=A. Cohn's irreducibility criterion}}

[[Category:Polynomials]]
[[Category:Theorems in algebra]]