Cohn's irreducibility criterion
Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in —that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.
The criterion is often stated as follows:
- If a prime number is expressed in base 10 as (where ) then the polynomial
- is irreducible in .
The theorem can be generalized to other bases as follows:
- Assume that is a natural number and is a polynomial such that . If is a prime number then is irreducible in .
The base-10 version of the theorem attributed to Cohn by Pólya and Szegő in one of their books[1] while the generalization to any base, 2 or greater, is due to Brillhart, Filaseta, and Odlyzko.[2]
In 2002, Ram Murty gave a simplified proof as well as some history of the theorem in a paper that is available online.[3]
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.
Historical notes
- Polya and Szegő gave their own generalization but it has many side conditions (on the locations of the roots, for instance)[citation needed] so it lacks the elegance of Brillhart's, Filaseta's, and Odlyzko's generalization.
- It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn, a student of Issai Schur who was awarded his PhD in Berlin in 1921.[4]
See also
References
- ^ George Pólya (1925). Aufgaben und Lehrsätze aus der Analysis, Bd 2. Springer, Berlin. OCLC 73165700.
{{cite book}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) English translation in: George Pólya (2004). Problems and theorems in analysis, volume 2. Vol. 2. Springer. p. 137. ISBN 3-540-63686-2.{{cite book}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Brillhart, John (1981). "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics. 33 (5): 1055–1059. doi:10.4153/CJM-1981-080-0.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Murty, Ram (2002). "Prime Numbers and Irreducible Polynomials". American Mathematical Monthly. 109 (5). The American Mathematical Monthly, Vol. 109, No. 5: 452–458. doi:10.2307/2695645. JSTOR 2695645. (dvi file)
- ^ Arthur Cohn's entry at the Mathematics Genealogy Project