Good afternoon. I don't speak English. Therefore, there may be errors in the text. I found a proof of the Abel-Ruffini theorem, without using the Galois theory. This is the original proof, translated into modern mathematical language with minor changes. Link to it: https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Rosen.pdf
I don't understand some features
- Section 5 proves that the first radical in a radical chain of extensions must be square. I don’t understand where the transposition and the square radical have to do with it.
- Why the radical expression of the square radical must necessarily have the form that is defined in section 3
Please explain without Galois theory
