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> That's the big disadvantage to the topological method I talk about (developed by Vladimir Arnold); it can only prove stuff about the general nth-degree formulas, whereas Galois theory can prove stuff about specific polynomials (e.g., that x^5 – x – 1 is unsolvable in radicals).

Can these topological methods be generalized to arbitrary fields?

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I'm a bit out of my depth here, but I think you can generalize it to differential fields, where you want to figure out e.g. which functions have a closed-form integral. A reference (the only reference?) would be Topological Galois theory: https://www.springer.com/us/book/9783642388705

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