Talk:Eisenstein's criterion

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User:Dysprosia, by all means add examples.

Charles Matthews 09:11, 4 Dec 2003 (UTC)

I'll probably give you some later tonight :) Dysprosia 09:12, 4 Dec 2003 (UTC)

Someone removed my edition of the article. In maths it is not important to put on the paper those things that does not lead a solution. Therefore I think it is better to say that only the case p=5. --Matikkapoika 16:04, 12 January 2006 (UTC)

In maths, it is sometimes more illustrative to demonstrate what doesn't work than to only adhere to what does. This aids comprehension on the reader's part. Now, you may be familiar understanding "those things that does not lead a solution" but some other reader may not. Dysprosia 22:45, 12 January 2006 (UTC)

Okay then. I just thought that if you want to know what is an Eisenstein criterion you know quite a lot mathematics beforehand. At least I heard Eisenstein's criterion first time in the university. Therefore I assumed that if you know what criterion says, you know also that mathematicians fails often before they can solve problems or prove new theorems. And therefore I assumed it is enought to give only the case p=5. Because if I would publish everything I think on my own, I would easily be the most prolific mathematician in the world! (Assuming everyone else publishes only valid results.) --Matikkapoika 00:21, 13 January 2006 (UTC)

It's not always easy for one to understand a new concept, even though one may have a large body of previous knowledge, precisely because it may be a new concept. Being deliberately terse may be fine in the body of an encyclopedic article, but for a worked example, it is best to be explicit. Dysprosia 01:11, 13 January 2006 (UTC)

I added a restriction in the general definition. If the criterion holds, then the polynomial is irreducible in F[x], where F is the field of fractions of D, and when the polynomial is primitive it is also irreducible over D[x]. This addition is important, since elements of D that are not units are considered irreducible elements of D[x]. Even in the integers, 2(x + 3) is the irreducible factorization of 2x + 6 (up to multiplication by 1 or -1), which matches the criterion for p = 3. However, since 2 is a unit in , 2x + 6 is irreducible in . Stolee 05:58, 16 February 2007 (UTC)

[edit] Examples

The change of variables in the examples is inconsistent. First it says to set x = y + a, and then redefines x as x + 3. The cyclotomic example says to set y = x + 1, which is the right way to do it. I think the section would make more sense if the first example was consistent with the second, and set y = x + a. Further, I think it should look like h(x) = h(y + 3) = y^2 + 7*y + 14 to be clearer and not redefine x.