Consider the mapping from the coefficients to the roots. Consider loops in the coefficient space. As you move around in loop and come to same point, the set of roots will come back to the same set but the individual roots could be permuted.
To show that the roots cannot be written in terms of nested radicals, one shows that loops formed by commutators have the property that sets defined by nested radicals return to their original positions. On the other hand we can find that the sequence of commutator subgroups for S5 stabilises at some point- so for any given length of nesting (commutators), non-trivial permutations can be induced on the roots by using loops formed out successive commutation of the given length. This shows that roots can’t have an expression as a finite nested radicals of continuous functions of the coefficients.
[The map corresponds to a Riemann surface given by the polynomial. And the permutations induced by the loops is the monodromy group. In this language, the above argument is saying that the mondromy is not solvable and nested radical expressions have solvable mondromy.]
https://projecteuclid.org/download/pdf_1/euclid.tmna/1471875703
A Neighbourhood Of Infinity 