I have a question regarding Arnold's proof of the insolvability of the quintic (https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf).
Here are my questions (but I'll give more context below):
(1) I don't understand what happens if the hypothetical quintic formula has division (the pdf linked above claims that division is allowed). Are we just guaranteed the the quintic formula never has division-by-zero?
(2) What if the loop in the function space results in one of the radicals inside the hypothetical quintic forumla becoming zero? Then we can't lift the path because we are passing through a branch point of the Riemann surface of $\sqrt[n]{z}$. Are we just guaranteed that the quintic formula never has $n$th-root-of-zero?
As far as I understand, here's the gist of the proof:
Consider the root-space $$\mathcal R=\{(r_1,\ldots,r_5)\in\mathbb{C}^5:r_i\neq r_j\text{ whenever }i\neq j\}$$ and the function-space $$\mathcal C=\{(a_0,\ldots,a_4)\in\mathbb{C}^5:x^5+a_4x^4+\cdots+a_1x+a_0\text{ is separable}\}.$$
Suppose that we have some quintic formula $E$ involving radicals. Consider a separable quintic $$x^5+a_4x^4+\cdots+a_1x+a_0=(x-r_1)\cdots(x-r_5).$$ Consider a permutation $\sigma\in S_5$ with no fixed points. Then we look at a continuous curve $\gamma_{\mathcal R}\colon[0,1]\to\mathcal R$ such that $\gamma_{\mathcal R}(0)=(r_1,\ldots,r_5)$ and $\gamma_{\mathcal R}(1)=(r_{\sigma(1)},\ldots,r_{\sigma(5)})$. This induces a continuous loop $\gamma_{\mathcal C}\colon[0,1]\to\mathcal C$ such that $\gamma_{\mathcal C}(0)=\gamma_{\mathcal C}(1)=(a_0,\ldots,a_4)$. The key idea seems to be to consider what happens when we plug $\gamma(t)$ into $E$. If $\sigma$ was chosen to be a sufficiently nested commutator (which exist in $S_5$), then plugging $\gamma(t)$ into $E$ results in a trivial loop. This means that $E$ was not able to follow the initial permutation $\sigma$ of the roots.