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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.

Gadi A
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  • It is the discrete Fourier transform. But the DFT is a purely algebraic notion; no analysis necessary. – Qiaochu Yuan Nov 20 '10 at 23:26
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    Could you please give a link to such a DFT proof? –  Nov 21 '10 at 00:47
  • @George: for the cubic formula it's given in the Wikipedia article http://en.wikipedia.org/wiki/Cubic_function#Lagrange.27s_method . For the quartic formula it should be similar but one might have to split into cases. – Qiaochu Yuan Nov 21 '10 at 01:56

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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

asmeurer
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Timothy Wagner
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    If the link for the topological proof does not work, try this one: https://www.tmna.ncu.pl/static/files/v16n2-02.pdf – user157227 Oct 20 '14 at 18:37