Talk:Lagrange's theorem (number theory)

Proof

Do we really need the proof written out like this? And what is the reliable source for it? Deltahedron (talk) 10:39, 21 October 2012 (UTC)

I admit the proof might be able to use some rewriting, but you don't need reliable sources for proofs if they're clear and any theorems referenced are sourceable. That's the beauty of math. :). This one, though, does not necessarily fulfill that requirement. I had to prove it using Euclid's Lemma for polynomials. David815 (talk) 22:46, 28 January 2015 (UTC)

The current proof (March 26 2020) is a nasty mess

I'm not an expert in formulating mathematical proofs, but having read many in my life, I can say, this one doesn't qualify.

Firstly the presentation is bad - at least a few line breaks to split up thoughts would help. It's hard to even follow the steps when reading this single pile of words.

Then it says "it's a standard fact" over and over again, that is not at all helpful. I mean, the whole message in Lagrange's theorem is also "a standard fact", so why bother even writing down a proof?

You could at least add a reference here and there leading to a place that will show how and why something is "a standard fact". Stating "standard facts" and presuming the reader has to know that, implies, that they're not the right audience for this article anyway, because they already know everything.

The original proof in the history may have been a bit simplified, but it was way better than what I'm reading right now. The current thing doesn't add up to a proof as far as I can tell and uses circular reasoning, like:

"Another standard fact is that a non-zero polynomial over a field has at most as many roots as its degree"

...which is sort of the essence of Lagrange's theorem. 80.138.118.155 (talk) 13:21, 26 March 2020 (UTC)

Revamped article

This article was terrible. The theorem statement was a little wrong ("an integer polynomial over $\mathbb {Z} /p$" means what exactly?), the proof was just awful: it was overcomplicated, it tried to be self-contained but referenced results that can prove the theorem more quickly and cleanly, it was a little wrong.... I looked at the proof in Tattersall's Elementary Number Theory in Nine Chapters, and it too is overcomplicated, unintuitive, and also slightly wrong (it ignores the f(x)=0 case), though at least it's understandably so--Tattersall bends over backwards to avoid discussing fields or polynomial rings. I mostly rewrote the page and added a new proof of my own. I don't have a source--this is very basic material any grad student in math should be able to come up with.

Also, the "Extensions" sentence about the same theorem and proof being true over arbitrary fields like R made absolutely no sense, so I deleted it. The theorem should have some generalization to number fields with essentially the same proof and tweaked assumptions; that might be of interest if anyone cares to hunt for one. 24.19.12.242 (talk) 00:48, 17 February 2014 (UTC)

@Wcherowi: I find your revert a little callous; it would have been nice if you had discussed it here first. You reverted my edit because it was OR, but the proof you've restored was itself OR (check the history for yourself). You've also ignored my numerous points about errors in the article. Surely you should have at least deleted the nonsensical "Extensions" section and changed the theorem statement to actually make sense after reverting?

Since the old version seems in every way worse, including being OR, I've reverted your (well-intentioned, though perhaps overly quick) revert. 24.19.12.242 (talk) 08:48, 17 February 2014 (UTC)

I agree that the "Extensions" section was poorly worded. I think it was intended to state that it is true in general that a polynomial over a general field has no more roots than its degree and that Lagrange is a special case of that: this is the approach taken by 24.19.12.242. I prefer the previous version as being a more elementary for an elementary formulation (as are the similar proofs of Tattersall and of Leveque). It might be better to have a comment on the relationship to unique factorisation for polynomials over a field and the validity of the theorem in general as another section. Deltahedron (talk) 17:57, 17 February 2014 (UTC)

I agree that the old version is far worse mathematically, but we are not writing journal articles here. By your own admission on this talk page, you replaced what was on the page by a proof of your own. This is clearly a violation of WP:OR and my duty as an editor is to remove it - no ifs, ands or buts, discussion was not required. I reverted, rather than fix the problems, because that was the easiest thing to do. I do not see why I should have to spend my time cleaning up your work –due to your not following (understanding?) Wikipedia policy. It is clear to me that you could improve this page (and it certainly needs it), but please do so while respecting the WP:Pillars. Find a proof that is to your liking, paraphrase it and put it into the article with references. I will let this stand for no more than a week before I revert/fix it again (but I can not speak for other editors). Bill Cherowitzo (talk) 18:09, 17 February 2014 (UTC)

That's rather interesting: there is no obligation to make the encyclopaedia worse and indeed WP:IAR mandates the opposite. The IP's proof is indeed valid, and I'm sure it can be sourced, but I prefer the proof of the two references, which is neither the earlier (unsourced as I pointed out over a year ago) nor the later version. There is no reason not to discuss that here in a calm manner: artificial deadlines rarely help discussion. Deltahedron (talk) 18:22, 17 February 2014 (UTC)

The earlier proof appears in Ivanov, Oleg A. (2009). Making Mathematics Come to Life: A Guide for Teachers and Students. American Mathematical Society. p. 74. ISBN 0-8218-8629-0. The later is quite close to Rose, H. E. (1995). A Course in Number Theory. Oxford science publications (reprint ed.). Oxford University Press. p. 41. ISBN 0-19-852376-9. Deltahedron (talk) 19:08, 17 February 2014 (UTC)

For what it's worth, I have no interest in writing up someone else's proof of such a basic fact; it takes literally minutes to come up with one. I feel Wcherowi is not listening to me and is being confrontational, which has turned me off from editing this article further, sorry. Thanks for your comments Deltahedron; it seems the old proof was simply unsourced rather than OR, so I can understand if you want to revert it ("lesser of two evils", I suppose?). Good luck all improving the page if you wish to spend time on it. 24.19.12.242 (talk) 00:22, 18 February 2014 (UTC)