Talk:Cohn's irreducibility criterion

Speedy deletion

Provides no context, but I think it's obvious that it's maths-related. Listed at Wikipedia:Missing_science_topics/Maths1, doesn't really make sense to go speedying articles which have been requested. --Cornflake pirate 11:33, 16 July 2006 (UTC)

Change subscripts and the title for this entry?

When this article refers to "n=2" it's not immediately clear whether it refers to the base or the degree of the polynomial. What makes Cohn's criterion cute is that you can convert a decimal number to a corresponding irreducible polynomial. Using 'b' as the base would be more clear. To implement this I'd like to change the displayed formulas to use the same subscripts as in Ram Murty's 2002 article. (Murty's article is more understandable than Brillhart et al). The other change I'd recommend is to change the name of the article to 'Cohn's irreducibility criterion' to make it easier to find in a Wikipedia search. EdJohnston 22:48, 18 August 2006 (UTC)

I have reinserted the statement of the criterion for a general base, but using b for the base instead of n, to avoid confusion. Also, note that the base 10 case requires each coefficient of the polynomial to be between 0 and 9 inclusive (i.e. a single digit in base 10). Without this restriction, the following would be a counterexample:

$10^{3}-9.10^{2}-9.10+1=11$ is prime but

$x^{3}-9x^{2}-9x+1=(x+1)(x^{2}-10x+1)$ is reducible in $\mathbb {Z} [x]$.

Gandalf61 13:34, 30 August 2006 (UTC)

Response

You are NOT using appropriate counter examples. You are violating the basic conditions of Cohn's criterion. The coefficients have to be integers between 0 and 9, inclusive. You are using integer coefficients between -9 and +9 inclusive. The original statement of the theorem uses the addition operator between polynomial terms, and states that the coefficients must be positive integers.

I originally came to this talk page today to ask if the coefficients could be negative. I do not know for sure that they cannot be in theory. But it is not stated or proven on this page; you, however, are just using this as such like its second nature. 64.134.140.53 (talk) 00:32, 29 September 2012 (UTC)

Found the original source

I agree with Gandalf's new wording. This is just a note to clarify which mathematician first gave each version of the theorem. (This is more detail than is needed in the article proper).

Library searches don't come up with any origin of Cohn's theorem except its appearance in Polya and Szego's book, published in 1925. (I looked it up in the original). They assert Cohn's theorem specifically for base 10 (vol. 2 page 137). They also include the condition that the leading digit be greater than one. I don't think it hurts to leave that condition out because the corresponding polynomial than just starts with the first non-zero digit. They do of course have the condition that all the coefficients range between 0 and 9 inclusive.

The unmodified generalization of Cohn's original rule to bases other than 10 was first given by Brillhart et al 1981. Polya and Szego (1925) have their own generalization but it has many side conditions (on the locations of the roots, for instance) so it fails to be completely Cohn-ian.

It's clear from context that this A. Cohn must be Arthur Cohn, a student of Issai Schur who got his PhD in 1921. He is listed in the Mathematics Genealogy Project. EdJohnston 20:52, 30 August 2006 (UTC)

Sources

Pólya, G., G. Szegö (1925). Aufgaben und Lehrsätze aus der Analysis, vol. 2. Springer, Berlin.

I think the authors are George Pólya and Gábor Szegő, however only the former article mentions the book, as Problems and Theorems in Analysis.

Aufgaben Und Lehrsatze Aus Der Analysis on Amazon.com shows publication as Dover (1945) and Springer; 4 edition (July 1, 1970). ISBN 3540054561 shows Springer; 4 edition (July 1, 1971). Is it ok to use the ISBN of the incorrect edition? John Vandenberg 12:51, 27 November 2006 (UTC)

I think it would be fine to provide the ISBN of a recent edition in English. In the present article, some results are identified by page number, so it would help to include the 1925 edition in the reference list. The 1925 is the only one I have actually seen. For this purpose, and if the book is too early to have an ISBN, an OCLC is ideal, since it specifies a particular edition. No harm in including a more recent edition as well. EdJohnston 00:58, 28 November 2006 (UTC)

Should we remove the second external link?

I suggest that we remove the second external link, a PDF of a 1998 paper by Nowicki and Swiatek. They don't seem to be aware of the 1981 result by Brillhart et al., and they present the base 2 case as an open question. (One that was solved 17 years prior to their publication). This has the potential of confusing our readers. I would not be against adding new external links to full-text papers about this problem that were germane and up-to-date. EdJohnston 01:38, 28 November 2006 (UTC)

The external pdf link was added before the article had any citations: [1]

With three referred sources, the pdf is no longer required for WP:CITE and the PlanetMath's explaination is easier to understand. John Vandenberg 02:47, 28 November 2006 (UTC)

Why remove references at all? The more references we have, the more confidence some people may be in the theorem. And not all references are worded the same, so some references may be easier or harder for some to follow. I think more is better than fewer. Of course, there does come a point of absurdity when you are just being redundant and posting far far too many references, and are just beating a dead horse at that point. Then again, as a pure mathematician, I dont see any legitimate reason for even one reference. All I need to see is just one valid proof, and to hell with citations. 75.172.58.58 (talk) —Preceding undated comment added 21:38, 4 October 2012 (UTC)

Dubious conjecture

"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."

If a polynomial with integer coefficients is irreducible, then its coefficients do have gcd 1. The author might have meant "whose values on integers have gcd 1", or "whose coordinates in the canonical basis of integer-valued polynomials are coprime" (otherwise his conjecture is obviously false), but then, the (unsourced) conjecture seems to me much weaker (hence − maybe − solved ?) than Bunyakovsky's. Anne Bauval (talk) 20:54, 2 January 2013 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Cohn's irreducibility criterion/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Could be linked in better with number theory, and possibly algebra. Donald Knuth also wrote about factoring polynomials, so cross-references are possible.

Last edited at 23:44, 19 April 2007 (UTC). Substituted at 01:53, 5 May 2016 (UTC)