Talk:Gauss's lemma (polynomials)

Article title[edit]

Shouldn't this be Gauss' lemma and not Gauss's lemma?

Google's top hits vote for "Gauss's". And I've usually heard it pronounced "gowses lemma" (not "gowse lemma"), if we follow that rule for the possessive s. iames 20:01, 3 May 2007 (UTC)[reply]

Section: "Proof of the lemma"[edit]

In the section "Proof of the lemma" where step 3 asserts the existence of a,b such that ag and bh are primitive, I believe that more justification is required. I'm working through this proof for the case R = Z and F = Q (in the notation of the article) and I can't find any good reason why a,b must be in R. It does follow that ab is a unit in R, which allows us to conclude that ag and bh multiply to f (up to units). In either case, I think the proof presented in this section needs some touching up.

a is a least common multiple of the denominators of the coefficients of g (when the coefficients are expressed in lowest terms). Algebraist 21:42, 22 April 2009 (UTC)[reply]

Hey Algebraist, I think you are wrong. What can you tell me about this: x^2 is obviously primitive. But you can factor it in f(x)=2/5*x and g(x)= 5/2*x. But you can't find an element a in R such that af(x) is primitive. —Preceding unsigned comment added by 187.39.190.134 (talk) 17:21, 23 January 2010 (UTC)[reply]

You're right. I've fixed the proof. Algebraist 18:38, 23 January 2010 (UTC)[reply]

One more thing. According to the way you stated de Lemma, you're not proving it by the contrapositive. —Preceding unsigned comment added by 187.39.190.134 (talk) 16:41, 25 January 2010 (UTC)[reply]

Proof of the lemma (again)[edit]

In the first proof of the primitivity I believe the sum for the highest degree monomial of the product contains only one term making the rest of the argument nonsense. A correct proof can be found here: [1] — Preceding unsigned comment added by 130.149.14.72 (talk • contribs) 13:25, 4 March 2015‎

It seems that you have not read the sentence until its end. The proof is correct. However, for clarification, I have edited the sentence by permuting some words. D.Lazard (talk) 15:56, 4 March 2015 (UTC)[reply]

New version of the article[edit]

A major rewrite of the article has been done recently. The old version has several issues; one is to be insufficiently sourced (tag at the top); this has been improved by the rewrite, and IMO this is its only improvement. The main other issue of the old version is to ignore the related Wikipedia articles and the main applications of the lemma (see below). This is worse in the new version.

The main difference between the two versions is to merge the section on GCD domains into the generalization to any domain. This seems WP:original synthesis. Moreover the new version changes the common definition of a primitive polynomial over a UFD by calling primitive the polynomials that were called comaximal in the old version (if $R=\mathbb {Z} [u,v],$ everybody calls primitive the polynomial $ux+v\in R[x],$ not the new version). Also, several assertions of the new version are mathematically dubious or, at least, insufficiently documented; for example the existence of a unique minimal principal ideal containing any finitely generated ideal, in a ring that is not an integral domain nor Noetherian, the existence. Overall, the main issue of this merged section is to be too technical for most readers, who come here from applications to UFD.

The issues of the old version that are kept by the new one are the following:

What is called irreducibility statement is not commonly called Gauss's lemma, as far as I know.

Otherwise, the following facts are lacking, and must appear in the article

The existence, in any GCD domain, of a factorization of every polynomial into primitive part and content, which is unique up to units, and is compatible with products.
The following generalization of the "irreducibility statement": If $R$ is a GCD domain with $F$ as field of fractions, every polynomial over $F$ is associated with a unique (up to the multiplication by a unit of $R$ ) primitive polynomial over $R$ , called its primitive part; this association is compatible with product, and therefore with irreducibility and factorization
The preceding is the basic tool for proving that a polynomial ring over a unique factorization domain is also a unique factorization domain.
This is also used in all modern algorithms for polynomial factorization and polynomial greatest common divisor.

The importance of these applications imply that a section on GCD domains must be kept.

For these reasons, I will restore the old version. D.Lazard (talk) 13:22, 19 January 2019 (UTC)[reply]

@D.Lazard: I don’t think reverting back to the old version is constructive; for example, the proofs in the new version are clearer (in my view) and fill some *gaps*; in fact, these gaps are the main reason for me for the major rewrite. Yes, there are some variations with the definitions; here I followed the standard textbooks (Eisenbud and Atiya and MacDonald). But I’m open to use a different source. If the main reason to reverting back to the old one is “mathematically dubious”, then I don’t think that’s the case here (and maybe that’s why you stroke-out that part). So, I will put the new version back but with some changes to address the issues. —- Taku (talk) 21:09, 19 January 2019 (UTC)[reply]

I guess “who come here from applications to UFD.” is the main point of the debate. Is it necessary to emphasize that particular case? The UFD case was also not emphasized in the old version. —- Taku (talk) 21:09, 19 January 2019 (UTC) In any case, I restored the main application (R UFD implies R[x]), which is indeed needed to be mentioned. (I didn't forget about it).[reply]

Finally, merging the general case with the UFD case isn't "WP:original synthesis."; this is the approach taken in Eisenbud. I think his approach (prove the general version first and then deduce the UFD case) is more streamlined and conceptually clearer and illuminating; or at least so for me. The editors are allowed for some latitude in the structure of the presentation, after all. -- Taku (talk) 21:25, 19 January 2019 (UTC)[reply]

Starting from the general version and then deducing the UFD case may be more streamlines for pure mathematicians. It is certainly not streamlined for mathematicians who are interested with applications of mathematics. And it is against Wikipedia guidelines (WP:TECHNICAL and MOS:MATH): Articles must be written with an increasing degree of technicality in order that the most general audience can understand the largest possible part of the article. In this case, everyone who know of GCD, polynomials and factorization should be able to understand everything, except the general case, which requires knowledge of ideals. Thus the general case must be kept at the end. On the other hand, if things are well presented, is not useful to present the UFD case separately from the integer case, as the proofs are identical, once it is said that Euclid's lemma and unique factorization are true. The statements and the proofs may be presented by starting: "If

Z

is the ring of integers or, more generally, a unique factorization domain, ..." I'll try to rewrite the article in this spirit.

On the other hand, I have never seen the second statement called Gauss's lemma. Moreover it is only one of the various consequences of the true Gauss's lemma. Therefore, I'll state it as a consequence. I'll edit sections one after the other. So, it may occur some incoherencies between edited sections and sections that are still not. D.Lazard (talk) 19:49, 25 January 2019 (UTC)[reply]

I'm in agreement that the first or early part of the article should be written for the readers "who know of GCD, polynomials and factorization" and no more. And, yes, the proof for the integer case work word-by-word for the UFD case. I think it was a separate section (and still is) since it is the case originally due to Gauss and is also the most well-known case. Maybe combining the integer and the UFD is ok; it could alienate a reader who knows Z[x] but not a UFD but a portion of such readers might be negligible enough...

I do disagree your claim that what is called the irreducibility statement here is not known as Gauss's lemma. That might be a region-dependent observation; my personal experience (in a graduate program in a US university) is that the fact "one can deduce irreducibility of a polynomial over Z from that over Q" is commonly referred to as Gauss's lemma. -- Taku (talk) 23:56, 25 January 2019 (UTC)[reply]