Talk:Eisenstein's criterion

Problem with how much to shift if polynomial is monic

The example with the polynomial h(x) = x^2 + x + 2 is given and a shift is needed in order to apply Eisenstein. How much should the shift be ? is 3 enough each time or does it depend on the polynomial. (I tried other shifts for the same polynomial and the test failed for them, However it worked for 3(given in the example on the wikipedia page) and 10 and 17 worked) Clearly they depend on the polynomial so how do you know how much to shift ? This should be added to the article Stefan.petrea (talk) 10:11, 1 February 2010 (UTC)

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 $\mathbb {Q}$, $2x+6$ is irreducible in $\mathbb {Q} [x]$. Stolee 05:58, 16 February 2007 (UTC)

A different generalisation

There's a generalisation of the criterion in Z (and probably over a general integral domain too) that says given a polynomial

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},$

if there's a prime number p such that p divides a_i for 0 ≤ i < k, and p² doesn't divide a₀ then f has an irreducible factor of degree at least k. Letting k = n gives back the usual criterion. There's no point cluttering maths articles with every generalisation of a result possible, though, so if someone thinks this is notable enough then feel free to add it to the main article. Chenxlee (talk) 19:44, 15 September 2008 (UTC)

That polynomial f should be monic, the way I've stated the result. Whoops. Chenxlee (talk) 19:53, 15 September 2008 (UTC)

Noteworthy

Yes, I find this very noteworthy. I do nott consider this an intuitively obvious result.There ought be some mention of it. I would add it save for the fact that I add nothing at all that I am not 100% sure of. Please add it yourself with either a proof or appropriate citations.

And dont you mean that the polynomial is irreducible for factors of degree k or less? Or for degree k or more?

Furthermore, doesnt this only matter up to k=floor(n/2) ? Polynomial factors with degree smaller than floor n/2 are accompanied by a second polynomial factor greater than floor n/2, which has degree n - k.

64.134.140.53 (talk) 02:09, 21 June 2012 (UTC)

Shifting and inverting algorithms

How does this tidbit effect the algorithms for shifting or reversing coefficients in order to "force" Eisenstein's Criterion.

64.134.140.53 (talk) 02:09, 21 June 2012 (UTC)

Problem with Generalization

The generalization given at the end of the article is wrong. Consider the domain $D=\mathbb {Z} [{\sqrt {8}}]$ and the polynomial $f(x)=x^{2}-2\in D[x]$. Now take the prime ideal $P=\langle 2,{\sqrt {8}}\rangle \subset D$. We have $2\in P$, but $2\notin P^{2}$, so this generalization of Eisenstein should apply... but it doesn't, since the field of fractions of D is $F=\mathbb {Q} [{\sqrt {2}}]$, and $f(x)$ is clearly reducible over F. I think we need D to be a unique factorization domain to fix it. Cpryby (talk) 23:52, 6 November 2008 (UTC)

Not irreducibility over Z

2X+6 satisfies the conditions with 3, but factorizes over Z as 2(X+3). So irreducibility is only over Q.80.99.11.161 (talk) 10:42, 21 October 2010 (UTC)

Change from f(x) to Q; improving this article

My recent change to this article, which among other things changed notation of polynomials from functional notations like f(x) to simple names (in capitals) like Q (I avoided the more natural P because of possible confusion with p for prime). This change, which was intended as a first step to improving parts of this article, was reverted with the summary "it is completely standard to denote a formal polynomial by f(x), no need to change notation". While I agree that many people and books almost always denote polynomials using "(x)" (thought very few are completely systematic; often people give up on this in expressions involving sums ans/or products of several polynomials, and this article is no exception), I don't think that this convention contributes positively to the readability or any other aspect of the current article. The only place where it is functional is to facilitate the notation for substituting x + 3 for x, as in h(x + 3), but even here this does not serve clarity: it masks the fact that there is an isomorphism argument that allows conclusions for h(x + 3) to be applied to h(x) as well. Also note that in the very same section this neat notational trick fails for the cyclotomic polynomial, because it is given by an explicit expression rather than by some name f(x).

I agree that my preference for using upper case instead of functional notation to distinguish polynomials from scalar is personal, though I am comforted by the fact that the polynomial article seems to adopt the same preference. I will defend (if challenged) that it is more consistent, but I'm not on a crusade to convince everybody to use it. But I maintain that adopting this convention for the current article (if done consistently) is certainly is no deterioration. If anybody thinks the opposite convention is really called for in this article, please explain on this talk page.

So I would please go on with some improvements to this article without being reverted. Simplifying the notation is just a first step to making it easier to understand and formulate. As a small justification, let me enumerate a few points that I think need improvement.

• The style needs to be more encyclopedic (less colloquial), and more precise
• The "basic proof" section is unclear as an argument (where for instance does it use that the one coefficient not divisible by p is in fact the leading coefficient?) and is hard to read, exactly because the distinction between polynomials and scalars is no longer clear.
• The "advanced explanation" section is not really an explanation (but I'm not competent to improve this part in other than superficial ways)
• There are numerous minor improvements possible (like removing the plural from "conditions" in the lead)

So I'll now turn back to the version I left, and make some of the improvements I intended. If a consensus arises that the functional notation is really better, I'll gladly turn that back afterwards, but I'd prefer not to be reverted globally. Marc van Leeuwen (talk) 08:38, 11 March 2011 (UTC)

I reverted you. First, I hadn't checked to make sure that you hadn't made changes other than the notational change you described in your edit summary. In the future, when you are doing making more than one type of change please indicate that in your edit summary. Second, if I come across an article where an editor has basically just changed from one notation to another with an edit summary describing how the new notation is better (when it's not clearly better), then I will revert that, because it often turns into that editor going through 10 other similar articles and making the same notation change. There are policies in wikipedia that basically say that if a convention is being used somewhere, you shouldn't just change it, not even to make things jive with other articles. Third, it's indeed typical in more advanced places to drop the (x) notation since it's redundant, and I generally do so and simply write f, g, h. Or I may even write P, Q. Or a, b, c. If you had spent time improving this article (say even just a few edits) and then had changed the notation, I don't think I would've reverted. At the time you reverted, there was no way to tell this is the first step in a larger program. Though I must say, I really don't like the notations G and H for polynomials (sure G without the presence of H, but both is jarring to me). In summary, it's not that I am of the firm belief that this article must use f(x) and such, it's simply that what seems to be a "drive-by" edit that completely changes a notational convention in an article to another one that is of about the same standard-ness is sketchy. All the other problems with the article you describe above are serious problems and I'd be very happy if you addressed them. RobHar (talk) 15:26, 11 March 2011 (UTC)

Regarding Gaussian Rational Roots

Does anyone know how or if Eisensteins Criterion says anything about the existence of Gaussian Rational roots? 64.134.140.53 (talk) 00:21, 21 June 2012 (UTC)

The Non-Primitive Caveat

The caveat needs to be elaborated and explaned in further detail, Im not sure that I am understanding it correctly.

This is my understanding: Even if Eisenstein's Criterion is met/passed and we have confirmed the non-existence of general rational roots... then we may still possibly have strictly prime roots if and only if the polynomial is not primitive.

64.134.140.53 (talk) 00:51, 21 June 2012 (UTC)

The criterion is about factorization, not about roots. The passage "It will also be irreducible over the integers, unless" can be illustrated as follows. Take an Eisenstein polynomial for p=5, say, for instance Q=2X³+10X²+25X+15. It is irreducible over the rational numbers by the criterion, and since this example is primitive it is also irreducible over the integers. However if we multiply by another prime, say 3, to obtain 6X³+30X²+75X+45, this still meets the Eisenstein criterion for p=5, the polynomial is still irreducible over the rational numbers, but it is reducible over the integers because it is equal to the product 3Q, and both 3 and Q are non-invertible factors over the integers. Speaking of reducibility over the rational numbers conveniently sweeps the factor 3 under the rug as invertible and therefore irrelevant. However this rather trivial case is the only reason to talk about the rationals; the criterion and its proof are really an integer affair. Marc van Leeuwen (talk) 13:05, 30 August 2012 (UTC)

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.

After the example of how to apply a shift it says that the shift is a ring automorphism, which is not true. — Preceding unsigned comment added by 131.111.221.163 (talk) 22:45, 12 June 2013 (UTC)

The substitution of x by x+a is an automorphism of the polynomial ring, mapping f(x) to g(x)=f(x+a). Such a substitution may also be viewed as a change of variable (y=x+a). The fact of changing or not the name of the variable is not really important, as the name of the variable in a polynomial is only a placeholder. Hover, in practice, it is safer to change the name, to avoid confusions. I have changed y into x in the example for a coherent notation in the section. D.Lazard (talk) 08:50, 13 June 2013 (UTC)