Talk:Fundamental theorem of algebra

message: " Shouldn't the theory itself be referenced? "  — Preceding unsigned comment added by OceanEngineerRI (talk • contribs) 22:28, 20 August 2013 (UTC)
thread_title: Reference

message: I've tried to fix the two links in the complex-analytic proof using Cauchy's theorem - I think the link was supposed to point to Cauchy's integral theorem rather than this disambiguation page. Can someone double-check this is the right place? Thanks.
--Schauspieler (talk) 09:53, 18 March 2012 (UTC)
thread_title:  Proofs section - "Cauchy's theorem" links to the disambiguation page rather than the specific theorem 

message: I think Abel's impossibility theorem should be worked into this article somehow, but I couldn't find a spot that I liked.  There are a couple of places in the article that say things like "it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degree is either 1 or 2" which can mislead people into thinking that you can always actually write down those coefficients/roots via some formula, but of course you can't. Abel's theorem says that there can be some roots that can only be approximated and not directly written down.  Wrs1864 (talk) 12:19, 26 April 2013 (UTC)
replies: text: Abels theorem can sensibly only be applied to the case where the coefficients have a finite representation. If the polynomial coefficients are general reals, then we are dealing with the impossibility of the solution of a non-realizable problem. Since we have no access to a faithful finite representation of the coefficients, all we can talk about are approximations of the polynomial and its roots, and their eventual limit. This is what Weierstraß was doing in 1889 when he invented the Durand-Kerner method in his homotopy proof of FTA.--LutzL (talk) 11:21, 10 March 2014 (UTC)
thread_title:  Abel's impossibility theorem 

message: I do not feel any of the current theorems satisfy how I think of the result.  This seems a special case of properties of continuous and smooth transformations of spheres into themselves.  Imagine an elastic sheet wrapped around a basketball.  Now, if we stretch and re-arrange the sheet, that's a transformation of the surface of the basketball into itself.  Now, the elastic naturally removes any wrinkles and folds, but we can get two layers by slicing the sheet, grabbing an edge, wrapping it all the way around the basketball, and then sewing it back to together.  We can do this process as times as we want, each time, getting one more layer of the elastic sheet around the ball.   Now, if we try to puncture the basketball with a nail, we'll have to go through as many layers as we wrapped around it.  So, if f(z) is the mapping represented by the elastic sheet, f(z) = a will have as many solutions as there were wrappings.
The key to the proof, then, is realizing that every polynomial over the complex numbers represents such a mapping of the sphere into itself, with the degree of the polynomial being the number of times the elastic sheet was wrapped around the basketball.  So, we need to show that the polynomial is continuous and doesn't fold the complex numbers (the core concept of an analytic function), and that the degree is the winding number.
So the theorem primarily a statement about the special properties of complex analysis.  — Preceding unsigned comment added by Uscitizenjason (talk • contribs) 03:10, 16 June 2013 (UTC)
replies: text: This is only another, but nice, visualization of the winding number proof.--LutzL (talk) 11:04, 10 March 2014 (UTC)
thread_title:  A constructive topological proof 

message: There seems to be an error in the topo proofs section.  It's stated:
  ... if a is a kth root of −p(z0)/ck and ...
Since z0 and ck are both constants this doesn't seem to make a lot of sense.  I think it's supposed to say
  ... if a is a kth root of −p(z0 + ta)/ck and ...
If I can convince myself that the proof makes sense that way and actually proves something I'll fix it but I'm still puzzling over it at this point (call me slow, whatever, I'm reading this page to try to grok the proof of the Fund. Theo. which I never really learned).  — Preceding unsigned comment added by Salaw (talk • contribs) 17:13, 9 March 2014 (UTC)
thread_title: Topological proofs section

message: The article lead currently states,
:"The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero."
By that argument, every root of every polynomial is complex by definition, since real roots have to be counted as complex.  Am I missing something basic here? SpinningSpark 23:36, 27 May 2018 (UTC)
replies: text: By definition, \R\subset\Complex, that is, real numbers are also complex numbers. Apparently, you are confused by the fact that in some contexts and in some poorly written texts, "complex" is used for "non-real complex". D.Lazard (talk) 08:13, 28 May 2018 (UTC)
replies: text: Please, do me the courtesy of assuming that I understand that.  So the theorem is just saying that all non-constant single variable polynomials have at least one root?  There is either no need to specify that the coefficients might be purely real, or else the statement should mention that both the coefficients and roots might be purely real. The confusion is not with me, but with the way this definition is worded. SpinningSpark 13:24, 28 May 2018 (UTC)
replies: text: I think that Prof. Lazard was just trying to take a stab at what you might find confusing in this statement of the theorem. The statement, as I see it, is just trying to be precise in specifying the conditions under which it is true. I have seen the theorem sloppily stated as "Every polynomial has a complex root" and I think the current lead is trying to avoid that kind of imprecision. Having real coefficients and/or real roots are just special cases and should not be part of the definition. The consequences of having real coefficients and/or real roots are brought up later in the article, but perhaps, since this is an important special case, some further mention might be made in the lead (but not in the paragraph that contains the statement of the theorem).--Bill Cherowitzo  (talk) 19:06, 28 May 2018 (UTC)
thread_title:  Does this definition mean anything at all? 

message: There is a neat proof with Lefschetz's theorem on fixed points. To say that some invertible n-by-n matrix A has an eigenvalue is the same as saying that it has a fixed point, seen as a map from CP^n-1 (complex projective plane) unto itself in the obvious way. Now this map is homotopic to the identity by connectedness of GL_n(C), so by Lefschetz we can conclude. It'd be great if someone would take the time to add it to the article, unfortunately idk much about wikipedia and I'd be afraid of messing something up. — Preceding unsigned comment added by 2a01:cb00:688:9a00:85f3:b104:5009:acfe (talk) 17:49, 20 April 2020 (UTC)
thread_title:  Topological proof 

message: The use of boldface in articles is not recommended, as was mentioned in the recent edit, which fixed the single case where the two topological proofs had to be separated from each other but weren't. In future edits, should the article be reformated using itemization or something similar?
Saolof (talk) 03:06, 21 September 2021 (UTC)
thread_title:  Formating of proof list?