For an oral exam I want to learn the proof of the Galois solvability criterion (Galois 1831).
I have already seen the proof online, I tried to understand it but it is incredibly long.
Could someone explain the proof idea to me by dividing the proof in some steps?
Since it is an oral exam I don't have to rewrite each passage. I just need to explain the key concepts of the proof and the different steps in short words.
The Galois solvability criterion:
Let $K$ be a field with $Char(K) = 0$. Let $f ∈ K[T]$.
Exactly then the equation $f(x) = 0$ can be solved by radicals if the group $Gal(f /K)$ is solvable.
Thank you for any help