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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?

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    $\begingroup$ What is your background in group theory and field theory? Without knowing this it may be difficult to give a satisfactory answer. $\endgroup$
    – Santana Afton
    Mar 1, 2019 at 16:01
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    $\begingroup$ (finite) group, group homomorphism, normal subgroup, group quotient, group action, cyclic group, simple group, solvable group, field, (finite) field extension, field automorphism - are just the most basic concepts needed beyond "this is a group" $\endgroup$
    – Hagen von Eitzen
    Mar 1, 2019 at 16:03
  • $\begingroup$ Galois showed the quintic was not generally solvable by radicals $\endgroup$
    – J. W. Tanner
    Mar 1, 2019 at 16:39
  • $\begingroup$ @JW Tanner - For clarity, I meant "solved" in the sense of a general proof of how they resolve - that some can be (and how, and which) and some can't be. $\endgroup$
    – Stilez
    Mar 1, 2019 at 19:32
  • $\begingroup$ @Sanata Afton - none. Much for reasons described in OP. But enough maths background more broadly, for a degree (even if outdated), so should be capable. $\endgroup$
    – Stilez
    Mar 1, 2019 at 19:34

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This is the question that got me into maths! I found John Fraleigh's "A first course in abstract algebra" to be a great source. It has plenty of text and lots of examples and exercises as well as historical notes. It also features a roadmap outlining which chapters you need to cover to understand the unsolvability of the quintic. Apart from the ones mentioned you should probably read 29-31, 48-51, and 53 as well to understand the field theory involved.

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