# Arnold's proof for the insolvability of the quintic

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.

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– user666369
Sep 7, 2019 at 15:57
• Thank you very much for this insightful book. I need to read it. Sep 7, 2019 at 17:17

In Exercise 1, take $$\alpha = 1/3$$, say. That's a cube root, ie a single nested radical.