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'''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 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]] | |||
Revision as of 17:53, 13 October 2013
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.
![{\displaystyle \mathbb {Z} [x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4da3ac703cc7721ebba91a53f6752de7157124)
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)Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang (dvi file) - ↑ Arthur Cohn's entry at the Mathematics Genealogy Project