[zyklu5]

There is indeed a deep connection between what is going on behind Arnold's proof and the classical Galois theory. But it needs quite a bit of sophistication to flesh out properly (not apparent in his famous lectures given to high school kids). There is a Galois theory for Riemann surfaces over algebraic functions where the coverings behave like fields do in the classical correspondence. If any one is interested, check out chapter 3 of Khovanskii's Galois Theory, Coverings and Riemann Surfaces.

[almostgotcaught]

i mean calling arnold's proof actually topological is probably a stretch (the "deep connection" you're talking about). it doesn't really use any topological facts about either the loops or the embedding space. continuity isn't really required i don't think? i should've said contiguity instead. it's just a very very nice model for the theory (in the sense of model theory) that lends itself to immediate visualization.