Is there an alternative proof of the abel-ruffini theorem?

Question

I'm just asking if there's a proof of the insolvability of the general quintic avoiding the uses of Galois correspondence and Galois extension.

Answer 1

This article may be to your liking. It still uses the concept of Galois groups (it's hard to see how one could avoid that), but does not use the Galois correspondence.

Answer 2

Here I have some related references.

Lerner, L.: Galois theory without abstract algebra. 2011

Goldmakher, L.: Arnold's elementary proof of the insolvability of the quintic. 2018

Zoladek, H.: The topological proof of Abel-Ruffini theorem. Topological Methods in Nonlinear Analysis 16 (2000) 253-265

Ramond, P.: Abel-Ruffini’s theorem: Complex but not complicated !. 2020

Yan Pan, Yuzhen Chen: A new proof of Abel-Ruffini theorem. 2020

Skopenkov: A short elementary proof of the insolvability of the equation of degree 5. 2020