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I have heard it mentioned as an interesting fact that the Fundamental Theorem of Algebra cannot be proven without some results from topology. I think I first heard this in my middle school math textbook, which said that Gauss searched unsuccessfully for a proof involving only algebra.

Yet, a necessary hypothesis for the FToA is the least-upper-bound property of the real numbers, which is itself a topological property; without this hypothesis the theorem is obviously not true ($x^2 - 2 = 0$). So, how could the FToA possibly not involve topology, and what was Gauss searching for?

Owen
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    No, that's wrong. FTA can be proven purely using complex analysis and algebra. Also, I wouldn't call the least upper bound property a topological property. It's just some fundamental analysis. – Balarka Sen Jun 19 '15 at 14:37
  • Also, this may help: link – TorsionSquid Jun 19 '15 at 14:55
  • @TorsionSquid Thank you, that link is helpful, and is really the exact same question as mine. I think I'll delete mine. – Owen Jun 19 '15 at 15:15
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    Dear @BalarkaSen : I dunno: one can argue fairly strongly by saying that the LUB axiom amounts to the topological completeness of the real numbers. This "no gaps" property is pretty solidly topological... Many parts of elementary analysis are purely topological, so I'm not sure where to draw the line. Regards – rschwieb Jun 19 '15 at 15:25
  • @TorsionSquid What exactly do you mean by "only needs the algebraic numbers"? I'm just not sure, so I'd appreciate the clarification. – rschwieb Jun 19 '15 at 15:29
  • @rschwieb Sure, if you're going to add analysis over $\Bbb R$ as the list of things needed in the proof of FTA, then there is no way to get away from that. I had in mind the winding numbers proof of FTA when I wrote that comment, and those involve pretty hard topological notions (indeed, quite distinguishable from standard analytic notions) like homotopy. – Balarka Sen Jun 19 '15 at 15:39
  • Related: http://math.stackexchange.com/questions/348646/fundamental-theorem-of-algebra-for-fields-other-than-bbbc-or-how-much-does, http://math.stackexchange.com/questions/165996/is-there-a-purely-algebraic-proof-of-the-fundamental-theorem-of-algebra. – Qiaochu Yuan Jun 19 '15 at 18:13
  • @rschwieb my bad, I'll delete the comment about algebraic numbers. – TorsionSquid Jun 19 '15 at 21:45

2 Answers2

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Yet, a necessary hypothesis for the FToA is the least-upper-bound property of the real numbers, which is itself a topological property; without this hypothesis the theorem is obviously not true

This is not true. The FTA is true for any real closed field in place of the real numbers, and most of these (for example, the field of real algebraic numbers) do not satisfy the least upper bound property. (One way to define a real closed field is a field which satisfies the same first-order statements as $\mathbb{R}$. The FTA is such a statement, but the LUB property is second-order.)

There is a proof of the FTA using Galois theory which reveals that it only depends on the following two facts about $\mathbb{R}$ (which I think are more or less equivalent to being real closed):

  • Every polynomial of odd degree has a root.
  • After adjoining $\sqrt{-1}$, every quadratic polynomial has a root.

Both of these are a corollary of the intermediate value theorem, but it's a nontrivial question whether it's necessary to appeal to this topological fact (since, again, there are real closed fields which are topologically very different from $\mathbb{R}$).

Qiaochu Yuan
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    This is exactly the explanation I was looking for. The key is that there are non-topological hypotheses that can be use (real-closed), but that it is unclear whether these hypotheses can be defined/used without reference to topology (ie, "same first-order statements as $\mathbb{R}$" references $\mathbb{R}$ whose definition involves LUB). This makes it very clear now. – Owen Jun 19 '15 at 18:37
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Gauss gave several proofs for the Fundamental Theorem of Algebra, see here, some of geometrical nature (Gauss-Bonnet Theorem), but no proof only using algebra. Indeed, all proofs known today involve some analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This fact has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.

Dietrich Burde
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  • "Indeed, all proofs known today involve some analysis, or at least the topological concept of continuity of real or complex functions." Do you have a reference for this fact? – RghtHndSd Jun 19 '15 at 15:19
  • "This fact has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra." I believe the question is asking for a reason why this is true. – RghtHndSd Jun 19 '15 at 15:20
  • For a reference, see here. The question asks why this isn't tautological. But the statement only involves algebra (well, and the complex numbers, of course). So it is not tautological to aks why there is no pure algebra proof. – Dietrich Burde Jun 19 '15 at 15:20