Lowest technology method to prove fundamental theory of galois theory in characteristic zero?

Question

I wish to learn the proof of the Abel-Ruffini theorem. I only care about what happens in characteristic zero. What's the method that gets me to a proof with the lowest level of technology?

I don't mind if the proof is long, if it illuminates what is actually going on. I've learnt a proof from Artin of the fundamental theory of galois theory, plus the fact that $S_n$ isn't solvable, but I don't feel I've gained any enlightenment from this.

I'm looking for a computational/mechanical/low level method to understand what's going on in the proof of Galois theory.

Answer 1

Leonard Dickson in some of his books (for instance (Modern) Algebraic Theories) does a nice job of presenting the Galois theory from first principles. It's interesting, if you want to see the way it was probably done by Galois itself.