Cohn's irreducibility criterion

Cohn's irreducibility criterion

Arthur Cohn's irreducibility criterion is a test to determine whether a polynomial is irreducible in \mathbb{Z}[x].

The criterion is often stated as follows:

If a prime number p is expressed in base 10 as p=a_m10^m+a_{m-1}10^{m-1}+\dots+a_110+a_0 (where 0\leq a_i\leq 9) then the polynomial
f(x)=a_mx^m+a_{m-1}x^{m-1}+\dots+a_1x+a_0
is irreducible in \mathbb{Z}[x].

The theorem can be generalized to other bases as follows:

Assume that b \ge 2 is a natural number and p(x)=a_kx^k+a_{k-1}x^{k-1}+\dots+a_1x+a_0 is a polynomial such that 0\leq a_i\leq b-1. If p(b) is a prime number then p(x) is irreducible in \mathbb{Z}[x].

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]

References

  1. ^ George Pólya; Gábor Szegő (1925). Aufgaben und Lehrsätze aus der Analysis, Bd 2. Springer, Berlin. OCLC 73165700.  English translation in: George Pólya; Gabor Szegö (2004). Problems and theorems in analysis, volume 2. 2. Springer. p. 137. ISBN 3-540-63686-2. 
  2. ^ Brillhart, John; Michael Filaseta, Andrew Odlyzko (1981). "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics 33 (5): 1055–1059. doi:10.4153/CJM-1981-080-0. 
  3. ^ Murty, Ram (2002). "Prime Numbers and Irreducible Polynomials". American Mathematical Monthly (The American Mathematical Monthly, Vol. 109, No. 5) 109 (5): 452–458. doi:10.2307/2695645. JSTOR 2695645. http://www.mast.queensu.ca/~murty/polya4.dvi.  (dvi file)
  4. ^ Arthur Cohn's entry at the Mathematics Genealogy Project

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Irreducible polynomial — In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non trivial factors in a given set. See also factorization. For any field F , the ring of polynomials with coefficients in F is… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”