TITLE: A different proof of the insolubility of the quintic?
QUESTION [12 upvotes]: I'm familiar with the "standard" proof using Galois theory that there is no general formula for solving an equation of fifth (or higher) degree using radicals (i.e. arithmetic and root-taking). However, now I'm wondering if other proofs of different nature were found (in particular ones relying on analysis rather than algebra). 
What sparked my interest was seeing a description of the solution of the 2nd, 3rd and 4th degree equations via something that looked like discrete Fourier transform.

REPLY [12 votes]: This is the content of the Abel-Ruffini theorem (whose proof predates Galois')
http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
A topological proof may be found in this paper.
https://www.tmna.ncu.pl/static/files/v16n2-02.pdf