Why is the observation that proof of the Fundamental Theorem of Algebra requires some topology not tautological?

Question

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?

Answer 1

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 $\mathbb{R}$).

Answer 2

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.