I am trying to understand Arnold's proof for the insolvability of the quintic from the manuscript:
https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf
which is actually well written. However, I am stumbling in Page 4 where the author says "By contrast, Exercise 1 implies that any combination of the four field operations, continuous functions, and a single nesting of radicals would produce a loop." and the subsequent Proposition 6.
Where does the single nesting of radicals come in Exercise 1? There is no nesting of radicals mentioned before this statement. Can anyone help? Thanks.
In Exercise 1, take $\alpha = 1/3$, say. That's a cube root, ie a single nested radical.