Talk:Fundamental theorem of algebra

From Wikipedia, the free encyclopedia

This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B+ Class	Top Importance	Field: Algebra
Consider nominating for Good Article status. Tompw 18:42, 7 October 2006 (UTC)

"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 Sep 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 Sep 2003 (UTC)

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

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

Remove the "less algebraic" argument. Not necessary. -wshun 21:34, 16 Sep 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 Jun 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 Oct 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 Dec 2004 (UTC)

Contents 1 Vagueness in winding number proof 2 factorizing 3 Generalizations 4 3 = 4? 5 Algebraic Proof

[edit] Vagueness in winding number proof

To me, a non-methematician, the winding number proof seems incomplete. Our article on winding number states that the winding number is a property of a given curve with respect to a given point. However, in the present article, we never indicate what point we're referring to. It also doesn't say why the existence of a zero of p(z) makes the statement any less absurd. Could someone familiar with this area please explain? --P3d0 17:49, Dec 19, 2004 (UTC)

The point referred to is the origin. (The article does say "counter-clockwise around 0".) In the proof, the loop is transformed in a continuous manner while the winding number changes from n to 0. In order for this to be a contradiction, there has to be no opportunity for the winding number to change as you transform the loop. But if p(z) = 0 somewhere, then the loop can at some stage pass through the origin, allowing the winding number to change. --Zundark 22:17, 19 Dec 2004 (UTC)

Ok, I made it "winding number with respect to zero". However, I think the proof is still unconvincing. What is the winding number of the constant circle |z|=0? Given what you just said, I presume it is zero. Are we sure of that? If so, it should be included in the proof; something like "... from the original circle to the constant circle |z|=0, whose winding number is 0, not n". I guess what I'm saying is that if we're going to claim something is absurd, it should be explicitly absurd. --P3d0 19:02, Dec 20, 2004 (UTC)

It's not talking about the winding number of |z|=0, but rather the winding number of p(z) when |z|=0. This is 0, as it's a non-zero constant function. --Zundark 20:48, 20 Dec 2004 (UTC)

Aha! Thanks for the reminder. --P3d0 04:04, Dec 21, 2004 (UTC)

Hang on... One of the premises of this proof is that if p has no zeros, then one can choose a value for z such that the zⁿ term dominates, and hence p will have a winding number of n with respect to zero. However, they then proceed to consider z values closer and closer to zero. Who says the winding number doesn't change once zⁿ no longer dominates? --P3d0 04:33, Dec 21, 2004 (UTC)

There is no possibility for it to change if the loop never crosses 0. This is sort of intuitively "obvious", especially for what we actually need in this proof (you can't take a loop encircling the origin and shrink it down to a point without hitting the origin). But it's not so easy to prove rigorously. If you really want to see all the details, you should consult an appropriate book. (If you don't have such a book you could download chapter 1 of Hatcher's book on algebraic topology. The Fundamental Theorem of Algebra is Theorem 1.8, but you will need to read the earlier parts of the chapter to understand the proof. The proof uses the fundamental group of the circle rather than winding numbers, but it amounts to the same thing - the elements of the fundamental group are essentially winding numbers.) --Zundark 14:42, 21 Dec 2004 (UTC)

Ah. And if it does cross 0, then the function has at least one zero. I think I get the picture now. Thanks for your patience. I think I may try to clarify some of these points in the article itself. --P3d0 17:13, Dec 21, 2004 (UTC)

[edit] factorizing

i read from somewhere something thats in direct contridiction with this article. in here it says any polynomial of degree n can be factorized as p(a)=(a-an)(a-an-1)...(a-a0) (sorry ima noob to wiki) either they are complex or real, but from that place it says it works only for n distinct real zeros. i think this article is correct but not sure, so can someone tell me the exact detail of this factorizing? also, i dont understand how the solution of general quintic is algebraic, like it cannot be solved exactly without using infinty series

I assume that in your undisclosed article there is referred to a specific method for factorising. Ofcourse it is possible that a certain method only works for real roots. This does not exclude the possibility to use other methods in case of complex roots. Bob.v.R 17:14, 15 September 2005 (UTC)

[edit] Generalizations

Can this theorem be generalized to (complex) matrix polynomials, or even to Banach algebras?

One generalization which has just been proven -- any field where all polynomials of prime degree have roots is algebraically closed. This improves on the proof given in the article, which requires roots for polynomials whose degree is 2 or odd. 24.149.215.73 16:15, 24 January 2006 (UTC) J. Shipman

It would be really worthwhile to include this generalization in the article, but your FOM posts are not enough to satisfy the WP:NOR criteria. Did you submit the result for publication in a journal? -- EJ 16:14, 8 January 2007 (UTC)

Generalisation to ANY other complex Banach algebra with unit fail at that first hurdle since a solution to aX-1=0 is an inverse to a and the only complex Banach algebra which is a field is C A Geek Tragedy 11:47, 18 February 2007 (UTC)

Since the FToA is equivalent to the statement that every complex matrix as an eigenvalue, one can view the nonemptiness of spectrum (for general Banach algebras) as a generalisation, especially since the standard way of proving it is the same as one of the proofs of the FT. Algebraist 21:36, 18 February 2007 (UTC)

OOO Nice :) A Geek Tragedy 19:15, 19 February 2007 (UTC)

[edit] 3 = 4?

for instance, he shows that the equation x⁴ = 4x − 3, although incomplete, has four solutions: 1, 1, − 1 + i√2, and − 1 − i√2.

This looks more like three true solutions to me... --Abdull 13:31, 9 June 2006 (UTC)

As is traditional, they're counted with multiplicity. That should be explicit though Algebraist 21:37, 18 February 2007 (UTC)

[edit] 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 ~~