I found the ideas in the book fascinating. The rough idea was:

• Show that the $n$th root operation allows for some "winding behaviour" on the complex plane.
• This winding behaviour of the $n$th root is controlled by $S_n$, since we are controlling how the different sheets of the riemann surface can be permuted.
• Show that by taking an $n$th root, we are only creating solvable groups.
• Show tha $S_5$ is not solvable.