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)

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.