The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize this jungle of proofs; this is enough proofs that we can start to discern some structure in the collection of proofs itself. The proofs seem to divide up into a few different categories; one attempt at naming these categories might be:
- Complex analysis proofs, e.g. the Liouville's theorem proof or the Rouche's theorem proof.
- Winding number proofs, e.g. Gauss's first proof (arguably), the Rouche's theorem proof (arguably), the $\pi_1$ proof in Hatcher.
- Topological proofs involving compactness, connectedness, openness, properness, e.g. this proof which is currently the top-voted one.
- Algebraic topology proofs different from the winding number proofs, e.g. the Lefschetz fixed point theorem proof, the proofs which consider $K^{\times}$ for $K$ a finite extension of $\mathbb{C}$.
- The inductive proof which reduces to the case of a polynomial of odd degree, which I think is equivalent to the Galois theory proof and which appears (to me) different from all the other proofs above.
Question: Which proofs of FTA are "essentially the same" and which are "essentially different"? Has anyone persuasively argued for a specific categorization of the proofs of FTA in the literature?
I think this is a rich test case for the general question of When are two proofs of the same theorem really different proofs?, and it seems like a natural enough question that I wouldn't be surprised if there was a survey paper out there that commented on this already. I'm not necessarily looking for a precise formalization of this notion of "sameness" or "difference" (I am skeptical that such a thing can exist even in principle), so informal but persuasive arguments are fine.
As a simple starting point, most proofs of the FTA don't generalize to real closed fields (since they rely on topological features of $\mathbb{R}$ or analytic features of $\mathbb{C}$), but the inductive / Galois theory proof does. So I think there's a strong case to be made that the inductive / Galois theory proof is "essentially different" from the others. (Although there's apparently a way to write down an algebraic version of the winding number entirely in the setting of real closed fields due to Eisermann, which can be used to prove FTA!) With these proofs we can be precise about exactly what facts about $\mathbb{R}$ are needed: namely, that every polynomial of odd degree has a root, and that every non-negative real has a square root. Other proofs make use of stronger facts about $\mathbb{R}$, e.g. connectedness.
I don't understand the complex analysis proofs in enough detail to tell which of them are "essentially the same" vs. different; all I can tell is that some of them are "essentially the winding number proof." On the other hand it seems to me that not all of the algebraic topology proofs are the same. The $\pi_1$ proof seems "essentially different" to me from the Lefschetz FPT proof, for example, although I don't know that I could argue for this persuasively, and maybe there is some nontrivial way to relate them! And the arguments using the fact that if $K$ is a finite extension of $\mathbb{C}$ then $K^{\times}$ is an abelian Lie group (which leads to a contradiction in several ways) seem to me "essentially different" again.
Alternatively, maybe the best we can ask for is a discussion of the "metric" structure of the space of proofs; I think we can at least persuasively argue that some of the proofs are "close" to each other, if not the same, and try to discern some clusters of nearby proofs, and so forth.
