Talk:Fundamental theorem of algebra

Is analysis required for proof?
"All proofs of the fundamental theorem of algebra involve some analysis, at the very least the concept of continuity of real or complex functions. This is unavoidable because the statement of the theorem depends on analysis; the sets of real and complex numbers are analytic objects. Some proofs also use differentiable or even analytic functions."


 * I don't think this makes much useful sense -- it's either trivial (if "some analysis" means "just the analysis to define C"), or untrue (meaning that the "because" doesn't hold above).
 * The algebraic complex numbers are algebraically closed, whereas the algrebraic real numbers are not. That difference is the essence of FTA, and the algebraic numbers are not inherently "analytic objects".  A proof for the algebraic complex numbers would be easy to extend to the whole of C, as C is just a transcendental extension of the algebraic complex numbers, and transcendental extensions are very simple and boring in that they introduce no non-trivial polynomial equalities, by definition.  Besides, FTA is a meaningful statement even if we define no metric at all on C (so there is no concept of a limit). (What sorts of proofs exist that the algebraic complex numbers are algebraically closed is a question of mathematical logic, though) 80.68.82.115 (talk) 23:49, 6 January 2008 (UTC)
 * The algebraic complex numbers are algebraically closed, whereas the algrebraic real numbers are not. That difference is the essence of FTA, and the algebraic numbers are not inherently "analytic objects".  A proof for the algebraic complex numbers would be easy to extend to the whole of C, as C is just a transcendental extension of the algebraic complex numbers, and transcendental extensions are very simple and boring in that they introduce no non-trivial polynomial equalities, by definition.  Besides, FTA is a meaningful statement even if we define no metric at all on C (so there is no concept of a limit). (What sorts of proofs exist that the algebraic complex numbers are algebraically closed is a question of mathematical logic, though) 80.68.82.115 (talk) 23:49, 6 January 2008 (UTC)


 * Now please define what a real field R is and under what conditions one considers a real algebraic field to be algebraically closed. The one definition I know involves the algebraic statement of the intermediate value theorem in the form that every polynomial with coefficients in R and odd degree has at least one root in R. Which is a sufficient setup to proceed via the Euler-Gauss proof (see Basu-Pollack-Roy). But that the "real" real numbers have this intermediate value property requires some basic real analysis.--LutzL (talk) 19:59, 7 January 2008 (UTC)


 * This quickly becomes a question not so much about which proofs are possible but about which proof steps one would consider "analysis", and how much of the reasoning back towards the axioms one wants to consider "part of" the proof. I tend to consider the "message" of the FTA to be one about the relation between R and C (essentially, that all we need to adjoin is i). Therefore, if one can derive the algebraic closedness of F[x]/(x²+1) assuming some purely algebraic properties of F, and then afterwards plug in F=R, then I would personally consider that to be an algebraic proof of the FTA even if one needed analysis to prove that R has the assumed properties in the first place. (Within reason, of course: it is easy to find a formal loophole in this description. However, the notion of what the "idea" in a proof is is an intuitive rather than a formal one in the first place). –Henning Makholm 19:09, 8 January 2008 (UTC)


 * I agree that the "because" from the sentence quoted above is untrue. Take, for instance, the assertion "For each $$x$$ &isin; R, $$-x^2$$ has a square root in C". It is about the set R of real numbers and about the set C of complex numbers. However, it can be proved without using any analysis.JCSantos (talk) 14:46, 8 January 2008 (UTC)


 * If anyone can cite a serious mathematician proving that any proof of the FTA must "involve some sort of completeness", then the sentence can be kept. Otherwise, it's just speculation from whoever wrote that sentence. It sounds too much like a theorem from mathematical logic to be kept with no reference. Quantum Knot (talk) 14:58, 18 December 2018 (UTC)
 * It is not only the proof that involves some notion of completeness, the statement of the theorem involves some notion of completeness. In fact, for proving anything, one requires a definition of what should be proven. The assertion "One knows what is C" is not sufficient here". So, for discussing this question, the FTA must be restated as the extension by a square root of –1 of the completion of the rationals is algebraically closed. As the concept of completion has been introduced, among other reasons, for insuring that $$\sqrt 2$$ is a real number, the proof that the equation $$x^2-2 = 0$$ has a real solution involves the concept of completion. A fortiori, the proof of the FTA involves this concept. I'll edit the article for explaining the disputed sentence with the example of $$\sqrt 2.$$ D.Lazard (talk) 16:09, 18 December 2018 (UTC)
 * The definition of complex numbers involves the word completeness, yes. I do not see why this implies that any proof of the FTA would involve analysis. The definition of real numbers involve analysis too, and yet no one would call linear algebra a part of "analysis"! Can you point out any mention of "completeness" in the proofs entitled "Topological proof", "Algebraic proof", and "Geometric proof"? This is Wikipedia, so unless you can find a source for this claim, then you should not write it here. Quantum Knot (talk) 08:50, 19 December 2018 (UTC)
 * Please provide an algebraic proof or a link to an algebraic proof that the equation $$x^2-2 =0$$ has a solution. About linear algebra: linear algebra is defined on any field, and every result of linear algebra remains true if the reals are replaced by any field (or any real closed field, for the most sophisticated results, such as Sylvester's law of inertia)
 * About the proofs given in the article: The topological proof starts by using that a lower bounded set of reals has a greatest lower bound; this is a definition of completeness. The topological proof starts with the intermediate value theorem, whose proof relies fundamentally on completeness. The "geometrical proof" is badly sourced (the source is a primary source). As it makes a heavy use of complex analysis, it is unclear whether it not circular (possible implicit use of FTA). In any case, it uses concepts of higher analysis that cannot been defined algebraically.
 * So, I'll restore my explanatory note, leaving your cn template. D.Lazard (talk) 09:55, 19 December 2018 (UTC)


 * I believe you are entirely missing the point, or I am. $$\mathbb{Q}[X]/(X^2-2)$$ is an algebraic object, not an analytical one. Yes, it is possible to define $$\sqrt{2}$$ analytically, but the algebraic definition is quite a bit more relevant here, don't you think? Perhaps I'm looking at it from a different perspective than you, but the essence of the theorem certainly seems to lie in $$\overline{\mathbb{Q}}$$ to me (and it's hardly controversial to say that the algebraic numbers are algebraic!). Anyway, it's the algebraic properties of $$\mathbb{C}$$ that are under consideration in the theorem, so at the very least the claim leading the "Proofs" section (that the theorem isn't even part of algebra) seems suspect to me. (And honestly, I think the modern perspective of these fields---namely, the very notion of algebraic completion---kind of supersedes the theorem.)
 * More importantly for the purposes of Wikipedia: since the sentences in question argue that the universally-accepted name is misleading, I don't see a justification for leaving it as-is in the article without a citation. (To the contrary, here is a "purely algebraic" statement and proof of the theorem.) GreatBigDot (talk) 02:22, 16 January 2019 (UTC)

Old discussion
"All proofs of the fundamental theorem necessarily involve some analysis,...

Agreed

"...since the very definition of the complex numbers requires the use of limits."

Is this statement true? Complex numbers are defined as ordered pairs of real numbers, with appropriate definitions for complex arithmetic operation. The crucial analytical property of the real numbers is Completeness, which can be defined in terms of the existence of Least Upper Bounds without reference to limits. Daran 04:06, 14 September 2003 (UTC)


 * Changed to "...since the construction of complex numbers from rational numbers requires the use of limits." Still not satisfactory, but more accurate. wshun 04:33, 14 September 2003 (UTC)


 * Changed again, and probably still not right. -- Daran 05:02, 14 September 2003 (UTC)


 * Maybe it would have been clearer to say that the construction of the complex numbers requires the use of limits, since every construction of R (infinite decimals, Cauchy sequences, Dedekind cuts, nested sequences of closed intervals, etc.) explicitly or implicitly makes use of limits. Proof that there exists a field satisfying the axiomatic characterization of R by the least upper bound principle requires the use of limits.


 * In any case, the current version of the article is genuinely wrong. The statement that the "complex rationals" (which is not standard terminology, by the way) satisfy the same algebraic properties as C is not merely wrong.  Juxtaposed with the statement that one is algebraically closed and the other is not, it is contradictory.  Michael Larsen 17:53, 14 September 2003 (UTC)


 * I meant the algebraic axioms, though of course you're quite right. I think I'm out of my depth here, and will bow out gracefully.  -- Daran 19:57, 14 September 2003 (UTC)


 * OK, I've fixed it up according to my lights. Take a look and see if you're satisfied with the new version.Michael Larsen


 * Er..., you talked me back into it. :-)
 * No I'm not. It is possible to prove the FTA from 1.  The axioms of the real numbers and 2.  The construction of the complex numbers from the reals.  It is not necessary to refer to any particular construction of the reals.  Moreover the axioms of the reals, specifically the LUB property can be stated without reference to limits.  Finally your statement that "Proof that there exists a field satisfying the axiomatic characterization of R by the least upper bound principle requires the use of limits." itself requires justification.
 * I'm not convinced that the statement "All proofs of the fundamental theorem necessarily involve the use of limits" is true. I agree with the statement "All proofs of the fundamental theorem necessarily involve some analysis" because I consider arguments involving the LUB property to be "analysis" even if they don't make use of limits.  (My old textbook on calculus defined "analysis" as "limits, etc.")  The definition given in mathematical analysis to which I've linked the article, is inadequate. -- Daran 03:37, 15 September 2003 (UTC)


 * Hmm, I'm happy with the article as it now stands, after your most recent revision. If you are, too, great!  If not, of course you can rewrite it.  If your new version annoys me, I may change it---or more likely, I'll just give up.   If I give up leaving a substantively incorrect article behind, I will consider it a failure of the wikipedia process.  I think wikipedia needs both professionals and amateur enthusiasts, but it may be that the two can't easily coexist...   Michael Larsen


 * I'm not happy with it, but I don't know how to improve it, so I guess I'll just have to remain unhappy. I think we can coexist... -- Daran 05:31, 15 September 2003 (UTC)


 * Since "use of limits", "algebra", "analysis" are not precise terms, the statement under contention is not mathematical or even metamathematical, but philosophical. Do all the arguments have an "analytic flavor" or a "limit-using flavor"?  I would say yes, and I think most mathematicians would agree.  But this is an area of legitimate disagreement, unlike the statement I removed in my last edit.  What I would like to do in my contributions to wikipedia, besides helping build up a database of accurate definitions and correct statements of theorems, is to convey some sense of mathematical culture, the kind of thing one imbibes by hanging around a math department common room listening to the grad students talking.  But I won't have much time to do this in the near future.   Michael Larsen


 * I have rewritten the sentence. I think "continuity of polynomials" is easier to understand than "limit" in construction of complex numbers, and it fits better for our three proofs in the article. -wshun 23:59, 15 September 2003 (UTC)


 * OK, I can live with the current version---I think the wording is a little awkward, but I have no mathematical or philosophical quarrel with it. Michael Larsen 03:40, 16 September 2003 (UTC)


 * I'm happy with that too. -- Daran 06:51, 16 September 2003 (UTC)


 * Remove the "less algebraic" argument. Not necessary. -wshun 21:34, 16 September 2003 (UTC)

(now considered something of a misnomer by many mathematicians)  - some mention why should probably be made? Dysprosia 06:47, 30 May 2004 (UTC)

Agreed! As is stands now, the reader may only guess why the remark is written there. My guess would be that 'many mathematicians' would nowadays replace 'algebra' by 'analysis'. But my guess may in fact be wrong. So, yes, a why would be a good addition here. Bob.v.R 05:24, 26 June 2004 (UTC)

I'm definitely not sure why: it was never called The Fundamental Axiom Of Algebra. It's fundamental in the sense that it validates much of the general analysis in algebra, the same way the fundamental theorem of calculus does there. Mark Hurd 15:43, 15 October 2004 (UTC)


 * On the page misnomer it says now: 'The Fundamental theorem of algebra can be proved from the axioms and is therefore not fundamental.' However, if it a theorem, it is obvious that it can be proven. So why can't it be a fundamental theorem? As it stands now, I would propose to delete the frase about the misnomer from this article. Bob.v.R 18:56, 11 December 2004 (UTC)

Algebraic Proof
The algebraic proof needs to state what exactly the induction hypothesis is, since there are different ways of expressing the theorem, which are non-equivalent when used as induction hypotheses. Also, the statement

So, using the induction hypothesis, qt has, at least, one real root; in other words, zi + zj + tzizj is real for two distinct elements i and j from {1,…,n}.

has too many commas and does not make much sense, since the induction hypothesis probably does not guarantee that a real root exists (unless something false is being proved, as there are many real polynomials without any real roots). —The preceding unsigned comment was added by 152.157.78.172 (talk • contribs) 15:35, 25 September 2006 (UTC)


 * I think you are right. I hope the new version is better. -- EJ 14:15, 8 January 2007 (UTC)

broken link: C. F. Gauss, “New Proof of the Theorem That Every Algebraic Rational Integral Function In One Variable can be Resolved into Real Factors of the First or the Second Degree”, 1799


 * Is this proof somehow circular? It supposes that the roots are all in $$\mathbb{C}\,$$. OTOH, I think that the proof could be made somehow more general than that, namely, that for any field F (is characteristic 0 required?) satisfying: (a) for every non-zero element x in F either $$\sqrt{x} \in F\,$$ or $$\sqrt{-x} \in F\,$$ (but not both) and (b) every odd-degree polynomial in F has a root in F, then F[i] is the algebraic closure of F. Albmont (talk) 16:36, 18 March 2009 (UTC)


 * Where do you see any circularity? Neither of the two proofs assumes the roots to be in C, you must have misread something. As for the generalization, the proofs (and indeed, the result) only works for real-closed fields (choose definition #3 from the list), i.e., on top of the conditions you mentioned, you also need to assume that F is formally real. (In both proofs, this condition shows up in the omitted proof that every number from F[i] has a square root.) — Emil J. 16:55, 18 March 2009 (UTC)


 * I got the circularity from here:
 * As mentioned above, it suffices to check the statement “every non-constant polynomial p(z) with real coefficients has a complex root”. This statement can be proved by induction on the greatest non-negative integer k such that 2k divides the degree n of p(z). Let a be the coefficient of zn in p(z) and let F be a splitting field of p(z) (seen as a polynomial with complex coefficients); in other words, the field F contains C and there are elements z1, z2, …, zn in F such that (...)
 * that is, the argument seems to imply that the splitting field of p(z) is C - or am I misreading something? As for the generalization, I guess that if F has a square root for (every element or its additive inverse) implies that F can be ordered, which implies that it's a formally real field. Albmont (talk) 20:16, 19 March 2009 (UTC)


 * First, you are misreading it. The text you emphasized just says that p has complex coefficients, it says nothing about its roots. In fact, the next sentence talks about F containing C and other stuff, which clearly suggests that F is bigger than C (at least a priori; the whole point of the proof is to show that, after all, F = C).
 * Second, it is not at all true that a field in which every number or its inverse has a square root (and −1 does not) can be ordered. Any finite field GF(q) with q ≡ −1 (mod 4) is a counterexample. By a more sophisticated construction, you can also find a counterexample where additionally every polynomial of odd degree has a root, and of characteristic 0. So no, you cannot omit the formally real condition. You can simplify it though: under the other condition it is equivalent to requiring that −1 is not a sum of two squares, you don't need an arbitrary finite number of them. — Emil J. 11:34, 20 March 2009 (UTC)


 * In fact, the generalization is mentioned in article real closed field: F is a formally real field such that every polynomial of odd degree with coefficients in F has at least one root in F, and for every element a of F there is b in F such that a = b2 or a = −b2. Maybe the algebraic proof of this fact could be placed in that article. Albmont (talk) 20:20, 19 March 2009 (UTC)


 * The article is not very clean on that, but I don't know how to change it without making it confusing. In the general case, the complex field should be taken as the quadratic extension C=R[i]=R[U]/(U2+1) of the real algebraic field R. Then the splitting field F is an algebra over C and the proof shows that it is of dimension one.--LutzL (talk) 10:09, 20 March 2009 (UTC)


 * Thank you all for making things clear for me. However, one bit of counterexample seems challenging: By a more sophisticated construction, you can also find a counterexample where additionally every polynomial of odd degree has a root, and of characteristic 0. I think this could be done by failing to have for some "positive" (an element is positive if it's non-zero and it has a square root) x and y a "negative" x + y. In other words, three "positive" elements x, y and z whose sum is zero. One such construction probably would have to take the Axiom of Choice and start with, for example, $$\mathbb{Q}[\sqrt{\pi}, \sqrt{e}, i \sqrt{\pi + e}]\,$$. Or am I missing some obvious construction? Albmont (talk) 13:22, 23 March 2009 (UTC)


 * Do you mean that you start with $$\mathbb Q(\sqrt\pi,\sqrt e,i\sqrt{\pi+e})$$, and find its maximal extension contained in C which does not include i using Zorn's lemma? Well, I don't see how to prove that $$i\notin\mathbb Q(\sqrt\pi,\sqrt e,i\sqrt{\pi+e})$$ (or equivalently, $$\sqrt{\pi+e}\notin\mathbb Q(\sqrt\pi,\sqrt e)$$) in the first place, as e and π are not known to be algebraically independent over Q. However, if you replace e and π with any algebraically independent pair of real numbers, then I think that this approach could work.


 * There are probably many different ways how to construct such a field, but what I had in mind was the following. First, fix any prime p ≡ −1 (mod 4), and let Kp be the union of all finite fields GF(pe) with odd e. (More precisely: construct the algebraic closure H of GF(p). Then H contains a unique isomorphic copy of each GF(pe), and the union of these copies for odd e is a subfield of H.) Then Kp has characteristic p, every polynomial of odd degree over Kp has a root, every number or its opposite has a square root, −1 has no square root, but −1 is the sum of two squares (indeed, every element of Kp is). Now, in order to construct a field with all these properties but with characteristic 0, we take an ultraproduct of all these Kp over a nonprincipal ultrafilter (or simply, we apply the compactness theorem), which works as there are infinitely many primes p ≡ −1 (mod 4). With a bit of care, the argument goes through in plain ZF without the axiom of choice (as the compactness theorem for countable theories does not need AC). — Emil J. 15:02, 23 March 2009 (UTC)
 * This ultraproduct is a very interesting construction... But there's no way to avoid the AC, since AC is equivalent to "every field has an algebraic closure" and "every filter is a subset of an ultrafilter". But my intuition breaks down even in the simplest cases; I can't figure out what is any ultraproduct of the $$\mathbb{Z}_p\,$$, just that it has characteristic 0 (maybe those ultraproducts aren't isomorphic; probably we can chose ultrafilters where -1 has a square root and ultrafilters where -1 doesn't). Albmont (talk) 12:21, 24 March 2009 (UTC)


 * First, AC is not equivalent to either of the two statements you mention, since both of them follow from the Boolean prime ideal theorem, which is provably weaker than AC (as shown by Halpern and Levy). In any case, this is irrelevant. You do not need the algebraic closure of every field, but only of a very specific field (a finite one), and in this particular case you can construct the algebraic closure explicitly. (More generally, the algebraic closure of any countable field can be constructed without AC.) The ultraproduct construction does indeed need some choice, but as I wrote, you only need the compactness theorem (the ultraproduct is only a way of doing a compactness argument in a more "algebraic" way), and the compactness theorem for countable theories is provable without AC. In fact, there is also another reason why the result is automatically provable without AC, if provable at all. The existence of the counterexample is equivalent to consistency of a particular recursively axiomatized first-order theory, hence it is an arithmetical $$\Pi^0_1$$-statement, and ZFC is conservative over ZF for such statements. For example, you can do the construction of the counterexample inside L, where AC always holds, and it will still work as a counterexample in the outer universe, as satisfaction of first-order formulas in a structure is absolute for transitive models of ZF.


 * As for properties of the ultraproduct, the key to that is Łoś's theorem. In particular, it implies that any first-order formula in the language of fields which holds in Kp for all but finitely many p also holds in $$\prod_pK_p/U\,$$. For example, that −1 has no square root is expressible by the formula $$\neg\exists x\,x^2+1=0$$, hence it holds in the ultraproduct. The other properties mentioned above (i.e., that every odd degree polynomial has a root, and every element is a sum of two squares) are also given by a set of first-order sentences, and are valid in all Kp, hence they are valid in the ultraproduct. That the characteristic is 0 is expressed by the set of sentences $$\underbrace{1+\dots+1}_{n\text{ times}}\ne0$$ for n > 0, and each one of these is valid in all but finitely many of the Kp (namely, it can only fail for p | n), hence the ultraproduct has characteristic 0. — Emil J. 14:12, 24 March 2009 (UTC)

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

" In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solving polynomial equations with real or complex coefficients " where was this statement taken from? it has no reference attached. —Preceding unsigned comment added by 189.221.32.18 (talk) 06:11, 17 October 2010 (UTC)

Proofs section - "Cauchy's theorem" links to the disambiguation page rather than the specific theorem
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)

Abel's impossibility theorem
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)


 * 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)

A constructive topological proof
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)


 * This is only another, but nice, visualization of the winding number proof.--LutzL (talk) 11:04, 10 March 2014 (UTC)

Topological proofs section
There seems to be an error in the topo proofs section. It's stated: 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 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)

''Actually what I wrote above doesn't make a lot of sense, clearly the reading should be something else. Anyhow after another cup of coffee maybe it'll all come clear. And this time I'll actuall sign my comment.'' Salaw (talk) 17:17, 9 March 2014 (UTC)

Never mind; I was being dense. I see it now. Sorry for the confusion. Salaw (talk) 17:27, 9 March 2014 (UTC)

Does this definition mean anything at all?
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)
 * 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)
 * 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)
 * 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)

Topological proof
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)

Formating of proof list?
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)