I’m studying maths at uni and I’ve just had a course in Galois Theory.
While I have understood the importance of the subject and while I can demonstrate most of the theorems, I can’t get my head around some basic ideas.
For example, what does it really mean for a group to be solvable?
I’ve “understood” Galois Theorem (about solvability by radicals and solvability of groups), but, for example, what’s missing in A_5? Why is A_4 solvable? Why are Dihedral Groups solvable? Why one even felt the need to give a name to a chain of groups whose factors are abelian?