|
|
| Line 1: |
Line 1: |
| '''Arthur 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 polynomials with integer coefficients.
| | The writer's title is Christy. Distributing manufacturing is exactly where her main income arrives from. Some time in the past she chose to reside in Alaska and her parents reside close by. To perform lacross is the factor I adore most of all.<br><br>Here is my weblog; accurate psychic predictions ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 http://cpacs.org]) |
| | |
| The criterion is often stated as follows:
| |
| :If a [[prime number]] <math>p</math> is expressed in [[base (exponentiation)|base]] 10 as <math>p=a_m10^m+a_{m-1}10^{m-1}+\cdots+a_110+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>.
| |
| | |
| 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_kx^k+a_{k-1}x^{k-1}+\cdots+a_1x+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>.
| |
| | |
| The base-10 version of the theorem attributed to Cohn by [[George Pólya|Pólya]] and [[Gábor Szegő|Szegő]] in one of their books<ref name=Polya>{{cite book | author = George Pólya | coauthors = Gábor Szegő | title = Aufgaben und Lehrsätze aus der Analysis, Bd 2 | year = 1925| publisher = Springer, Berlin | oclc = 73165700 }} English translation in: {{cite book | author= George Pólya | coauthors = Gabor Szegö | title = Problems and theorems in analysis, volume 2 | publisher=Springer | year=2004 | volume=2 | isbn=3-540-63686-2 | page=137 }}</ref> while the generalization to any base, 2 or greater, is due to Brillhart, [[Michael Filaseta|Filaseta]], and [[Andrew Odlyzko|Odlyzko]].<ref name=Brillhart>{{cite journal | last = Brillhart | first = John | authorlink = John Brillhart| coauthors = [[Michael Filaseta]], [[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 }}</ref>
| |
| | |
| In 2002, [[Ram Murty]] gave a simplified proof as well as some history of the theorem in a paper that is available online.<ref>{{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/polya4.dvi | doi = 10.2307/2695645 | jstor = 2695645 | publisher = The American Mathematical Monthly, Vol. 109, No. 5 }} (dvi file)</ref>
| |
| | |
| 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|date=September 2007}} 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.<ref>[http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=17963 Arthur Cohn's entry at the Mathematics Genealogy Project]</ref>
| |
| | |
| ==See also==
| |
| | |
| *[[Eisenstein's criterion]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==External links==
| |
| *{{planetmath reference|id=6194|title=A. Cohn's irreducibility criterion}}
| |
| | |
| [[Category:Polynomials]]
| |
| [[Category:Theorems in algebra]]
| |
The writer's title is Christy. Distributing manufacturing is exactly where her main income arrives from. Some time in the past she chose to reside in Alaska and her parents reside close by. To perform lacross is the factor I adore most of all.
Here is my weblog; accurate psychic predictions (http://cpacs.org)