I studied maths to degree level many years ago, but never got a "gut feel" for any of the "pure maths" side, because the way it was taught didn't fit at all with how I learn. I focused on numerical analysis and methods instead. I understood the basic concept of "this is a group", but after that it rapidly nosedived (for me) into new terms unconnected with anything that bound them together and gave them purpose and utility. A shame, but many years ago and I've forgotten most of what I knew.
But I've always wanted to get a sense of what I missed, a way to make those concepts real, by attaching them to an concrete and real problem, that's not easily solvable using anything I know, but is solvable with those tools.
I remember reading that Galois was the first to realise that small, carefully constructed groups could be used to crack specific problems, and it occurs to me that Galois' approach to the quintic might be ideal to digest and learn from - a concrete task likely to be complex enough to need such tools, simple enough to dig into and understand exactly where Galois was going with it, and the details needed to understand it.
So I think I've picked my 'problem' or it's picked me.
I have no idea how complex it'll be, so a really good walkthrough will be helpful - ill have to look for resources. But what do I actually need to comprehend, for the task? (A bit like a route map not just a bare skills list). And what tips would help before starting?
Relatedly - how did he even come u9 with the idea that there could be such an approach?