The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result was keep. — CharlotteWebb 04:01, 1 February 2007 (UTC)[reply]

• I withdraw this nomination (not that it matters). See comments below. --N Shar 00:10, 31 January 2007 (UTC)[reply]

I have no idea what to do with this article. It was created at Conmplex Conjugate root Theorem, so I moved it to its current location. I also cleaned it up. The only problem is that the Google test totally fails, and the theorem is in any case a corollary of the fundamental theorem of algebra as stated in the article and in Polynomial. So we have a few options:

• Keep at Complex conjugate root theorem
• Move to some other title
• Redirect (or merge and redirect) to some other article
• Delete entirely

I abstain, because I can't decide. N Shar 01:35, 29 January 2007 (UTC)[reply]

• Merge and Redirect to Fundamental theorem of algebra looks like a stub that would best work there (but I might be wrong not a math major). If it grows it can have its own page back. Jeepday 02:00, 29 January 2007 (UTC)[reply]

Change Vote to Keep the article is no longer a stub and the nomination for AfD has been withdrawn. Jeepday 03:05, 31 January 2007 (UTC)[reply]

• Merge and redirect per above - ∅ (∅), 02:28, 29 January 2007 (UTC) Keep as per Michael Hardy's comment below - ∅ (∅), 10:02, 30 January 2007 (UTC)[reply]
• Merge and redirect - the above article has a section for corollaries. Why does this need its own article, given that it's a highly straight-forward adaptation? --Haemo 03:34, 29 January 2007 (UTC)[reply]
• Merge and redirect. Since when do corollaries merit articles on their own? --Wafulz 03:53, 29 January 2007 (UTC)[reply]
  • Comment. I've said this elsewhere, but lest the above comment mislead anyone I'm saying it here as well:

(1) The proposition in NOT a corollary of the fundamental theorem of algebra.

(2) Even if it, were, the fact that it is such a corollary would be far from the most important fact about it, perhaps harly even worth mentioning in this article.

(3) "Since when do corollaries merit articles on their own?" is colossally silly. Silly, silly, silly, silly. Whether a topic warrants an article has nothing to do with whether it is or is not a corollary of something else. Can anyone cite ANY article that got deleted because it's a corollary of something else? Michael Hardy 17:08, 30 January 2007 (UTC)[reply]

• Weak keep (maybe to change to "strong keep" later?) This proposition is much weaker than the fundamental theorem of algebra. You don't need the fundamental theorem of algebra to prove it. So even if it is a corollary to that theorem, it silly to regard that as being an important fact about it. To say "Since when do corollaries merit articles on there own?" is also silly; that depends very much on context. Fundamental theorem of algebra is not the right place to merge it into if it is to be merged. Really, the comments above are silly, silly, silly, silly. If you believe, just because this article says so, that the main thing to be said about it is that it's a corollary of a theorem not needed to prove this much simpler theorem, then you are gullible and I can offer you a really great deal on some real estate in Florida. Michael Hardy 23:34, 29 January 2007 (UTC)[reply]
  • Comment I've deleted the silly comment that this theorem is a corollary of the fundamental theorem of algebra. The fact that the proof given in the article does not rely on the fundamental theorem of algebra makes that comment even sillier. But would any of the above commentators explain to me how this simple can be regarded as a corollary of the fundamental theorem of algebra? Michael Hardy 23:39, 29 January 2007 (UTC)[reply]
• Merge into complex conjugate. Hmm, on further thought just redirect, as complex conjugate already does tell what needs to be told in as many words as it needs. Michael Hardy is right that this cannot reasonably be described as a corollary of the fundamental theorem (I would strengthen that to denying that the fundamental theorem of algebra even implies it any more than the quadratic reciprocity theorem implies 2+2=4, namely in the trivial sense that any two true statements imply each other). However, I'm unconvinced that this theorem merits an article in itself. Is there anything encyclopedic to say about it other than to state it? I wouldn't consider proving it encyclopedic. Henning Makholm 00:42, 30 January 2007 (UTC)[reply]
  • Comment If the proof is deleted, there's still something to include beyond the bare statement: some simple corollaries of it and some examples of its use. I'll look at it again. Michael Hardy 02:25, 30 January 2007 (UTC)[reply]
• Keep. This page may be small, but it states a frequently used fact that deserves to stand on its own. The fundamental theorem of algebra asserts that every polynomial in C[x] has a root in C. The theorem under discussion does not assert the existence of a root, but merely the fact that any complex root of a real polynomial, if it exists, is accompanied by its complex conjugate as (another) root. As a corollary, it asserts that a real polynomial of odd degree has a real root, perhaps because the complex roots occur in pairs. Without checking the reference I do not know how it is proved there, but it is common to use the fundamental theorem of algebra to show that a polynomial of degree n has n roots; still, that dependency may be non-essential. It is true that the brief complex conjugate article devotes one sentence to this theorem, but if I were to cite this theorem I predict my readers would find a devoted article more helpful. --KSmrq^T 02:05, 30 January 2007 (UTC)[reply]

  Comment. I've been looking at a lot of random articles lately, and I find that most are brief mentions of topics verging on trivia. On reflection, I see no problem with that. This mathematics article covers an extremely important practical fact, far more worthy of attention than most of the random topics. Maybe I'm becoming a Wikipedia "inclusionist". :-) --KSmrq^T 02:05, 30 January 2007 (UTC)[reply]

• Keep as procedural, as no valid reason for deletion is given. If a valid reason is proposed, please relist —siroχo 08:33, 30 January 2007 (UTC)[reply]
• Keep - Now that the misleading stament about it being a simple corrollary has been clarified, it is a decent small article that may be helpful, is well referenced and linked to form and to others. Tikiwont 09:15, 30 January 2007 (UTC)[reply]
• Keep, possible rename The page expands a lot more on the theorem than complex conjugate does. What we really need is a source to find out what the theorem is called (if anything). (Looking for a source, I found [1], which demonstrates that the theorem is real, but doesn't give it a name; I suspect most maths textbooks (all obvious sources for this) will do the same.) --ais523 11:46, 30 January 2007 (UTC)
• Merge as a section into complex conjugate. This result is an important aspect of algebra and is probably taught in every basic course on complex numbers. However, I've not seen a textbook give it a name, probably because the result is almost always proven (in basic complex number courses at least) as a corollary of the Fundamental theorem (and corollaries are rarely named). Inasmuch as there no name to give to this thing, but nonetheless the information must be preserved, complex conjugate seems the best place to put it. Zunaid©® 14:36, 30 January 2007 (UTC)[reply]
  • Comment Here the claim that this result is a corollary of the fundamental theorem of algebra surfaces again. How on earth would you even use the fundamental theorem of algebra to prove this? Note that the result as stated in the article say that the roots pair up as counted with with multiplicity, simply that the set of roots is symmetric around the real axis. Henning Makholm 15:24, 30 January 2007 (UTC)[reply]
• Keep. In the future, this page could maybe be renamed into Properties of polynomial roots. (This would solve the google test and the somewhat clumsy name. See above comment by ais523.) In addition to the complex conjugate property, such a page could also disucss: 1) how roots depend continuously but not differentiably on the coefficients, 2) bounds on the roots in terms of the coefficients. There are probably many other properties of polynomials that merit a discussion. Haseldon 18:55, 30 January 2007 (UTC)[reply]
• Comment The "google test" DOES NOT FAIL!. Try it and see. Either with the whole phrase in quotes or without the quotes. Even without the quotes, the first pages of google results are on the same topic. It is unreasonable to do the google test with the quotes anyway, since this is not a case where verbatim identity of names is to be expected. Michael Hardy 19:50, 30 January 2007 (UTC)[reply]
  • Whoa there! No need to shout. - ∅ (∅), 00:03, 31 January 2007 (UTC)[reply]
    • No problem. I deserve it, really. --N Shar 00:10, 31 January 2007 (UTC)[reply]
• Withdraw nom. Michael Hardy is right, this is not a corollary of the FTA. I was frankly a bit surprised to see the assertion that it was, and I was a bit too willing to trust that assertion. The problem is that I'm not a good enough mathematician to be able to say that just because I can't see how it's a corollary, it isn't. Also, I grabbed the nearest algebra text (which was a high-school level text) and checked, and found this theorem listed, unnamed, as a corollary of the FTA, but without a satisfactory proof. These are the factors that influenced that statement, but clearly calling it a "corollary" is silly. The article has, in any case, been much improved by Michael Hardy and others, and sources have been provided which (presumably) verify that the name is in widespread use, which was what I was getting at with the Google hits test. --N Shar 00:10, 31 January 2007 (UTC)[reply]
• Keep Well written article on an important concept. Oleg Alexandrov (talk) 03:29, 31 January 2007 (UTC)[reply]

The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.