12
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ It is the discrete Fourier transform. But the DFT is a purely algebraic notion; no analysis necessary. $\endgroup$ – Qiaochu Yuan Nov 20 '10 at 23:26
  • 1
    $\begingroup$ Could you please give a link to such a DFT proof? $\endgroup$ – user1119 Nov 21 '10 at 0:47
  • $\begingroup$ @George: for the cubic formula it's given in the Wikipedia article 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. $\endgroup$ – Qiaochu Yuan Nov 21 '10 at 1:56
12
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.