Talk:Cohn's irreducibility criterion

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Mathematics rating:	B Class	Mid Priority	Field: Number theory
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Could be linked in better with number theory, and possibly algebra. Donald Knuth also wrote about factoring polynomials, so cross-references are possible.

[edit] 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)

[edit] 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:

10³ − 9.10² − 9.10 + 1 = 11 is prime but

x³ − 9x² − 9x + 1 = (x + 1)(x² − 10x + 1) is reducible in .

Gandalf61 13:34, 30 August 2006 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)