Cohn's irreducibility criterion

The criterion is often stated as follows:

:If a prime number $p$ is expressed in base 10 as $p\; =\; a\_m\; 10^m\; +\; a\_\{m-1\}\; 10^\{m-1\}\; +\backslash cdots+\; a\_1\; 10\; +\; a\_0$ (where $0\backslash leq\; a\_i\backslash leq\; 9$) then the polynomial

::$f(x)=a\_mx^m+a\_\{m-1\}x^\{m-1\}+\backslash cdots+a\_1x+a\_0$

:is irreducible in $\backslash mathbb\{Z\}[x]$.

The theorem can be generalized to other bases as follows:

:Assume that $b\; \backslash ge\; 2$ is a natural number and $p(x)\; =\; a\_k\; x^k\; +\; a\_\{k-1\}\; x^\{k-1\}\; +\backslash cdots+\; a\_1\; x\; +\; a\_0$ is a polynomial such that $0\backslash leq\; a\_i\; \backslash leq\; b-1$. If $p(b)$ is a prime number then $p(x)$ is irreducible in $\backslash mathbb\{Z\}[x]$.

The base 10 version of the theorem is attributed to Cohn by Pólya and Szegő in Problems and Theorems in Analysis{{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}} while the generalization to any base b is due to Brillhart, Filaseta, and Odlyzko.{{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 }} 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 Frederick William University in 1921.[http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=17963 Arthur Cohn's entry at the Mathematics Genealogy Project]{{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}}

A further generalization of the theorem allowing coefficients larger than digits was given by Filaseta and Gross.{{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 }} In particular, let $f(x)$ be a polynomial with non-negative integer coefficients such that $f(10)$ is prime. If all coefficients are $\backslash leq$ 49598666989151226098104244512918, then $f(x)$ is irreducible over $\backslash mathbb\{Z\}[x]$. 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.{{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 }}

An analogue of the theorem also holds for algebraic function fields over finite fields.

The 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.{{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)

• Eisenstein's criterion
• Perron's irreducibility criterion

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

Category:Polynomials

Category:Theorems in algebra