# Arnold's Topological Proof for the Insolvability of the Quintic

It seems similar questions have been asked about these notes by Leo Goldmakher (linked below) which give a topological proof by Arnold of the insolvability of the quintic by radicals. I think I understand everything except for how to prove exercise 1 on p. 4.

For a bit of context, $$\mathcal{F}_n$$ is used to denote the space of complex polynomials (with leading coefficient equal to 1) without repeated roots. So a polynomial is of the form $$z^n+a_{n-1}z^{n-1}+...+a_1z + a_0$$; it is determined by the coefficients and so such polynomials may be regarded as a subset of $$\mathbb{C}^n$$. The roots may be regarded as tuples in $$\mathbb{C}^n$$.

For example, if we take $$n=2$$, we have some quadratics and there is a quadratic formula which gives us roots. However, if you move the coefficients in a large loop (so take a loop in $$\mathcal{F}_2$$) and keep track of the roots, it's possible to permute the roots. This means that the quadratic formula does not always map a loop of $$\mathcal{F}_2$$ to a loop in $$\mathbb{C}^2$$. This is, of course, because square roots aren't really continuous functions on $$\mathbb{C}$$. We need branch cuts and the theory of Riemann surfaces originates at this point.

Now, here is the exercise.

Exercise 1: Suppose $$\gamma_1$$ and $$\gamma_2$$ are two loops based at the same point in $$\mathcal{F}_n$$, and pick any continuous function $$f: \mathcal{F}_n \to \mathbb{C}$$. Then for any $$\alpha \in \mathbb{Q}$$ the image of $$f(p)^\alpha$$ as $$p$$ traverses the commutator loop $$[\gamma_1, \gamma_2]$$ is also a loop in $$\mathbb{C}$$.

This exercise seems to be saying there is something special about commutators; I don't think the statement is true for arbitrary loops and it seems to me that the $$\alpha$$ exponent should give rise to the same issues as we have with square roots not being continuous. So I'm somewhat at a lost on how to prove the statement though I imagine that it should be rather elementary. Any help is appreciated.

https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

• There's a really excellent video about Arnold's proof here (by math.SE user not all wrong, by the way). Maybe that can help you sort it out? Oct 16, 2021 at 13:35
• @HansLundmark Thanks for sharing the video. I am still curious about this particular exercise though and I don't think the video discusses it. Oct 17, 2021 at 15:05
• @inkievoyd: Start at 22:47 of the video; the climax (and essentially a restatement of the counting argument you give below) is at 26:53 or so. The video calls the function $r$ instead of $f$, but the idea seems the same. Dec 17, 2021 at 0:21

I think one argument goes like this. We can just focus on $$\alpha = 1/n$$ and consider loops $$\gamma_1,\gamma_2$$ such that $$f(\gamma_1),f(\gamma_2)$$ are loops around zero. Otherwise, the $$1/n$$ isn't an issue.
WLOG, let's suppose that we may begin the image of the loops at $$1 \in \mathbb{C}$$. Then $$f(\gamma_1)^{1/n}$$ must have endpoint on some $$n$$th root of unity, say $$e^{2\pi ki/n}$$ for some $$0 \leq k \leq n-1$$. Similarly, $$f(\gamma_2)^{1/n}$$ ends at some other $$n$$th root of unity, say $$e^{2\pi \ell i/n}$$. Then the path traced out by $$f([\gamma_1,\gamma_2])^{1/n}$$ goes $$k$$ steps, then $$\ell$$ steps, then back $$k$$, then back $$\ell$$. This brings us back to $$1 \in \mathbb{C}$$ and hence, is a loop..