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I would argue that turning the question about the quintic into the question "which polynomials can be solved by radicals" is not a simplification at all. To develop Galois theory is easy now, in retrospect, but back then it was a tremendous achievement and involved amazingly deep insights and innovative thinking.
I also don't quite understand, what parts of your post constitute an answer to my question. E.g., I hope that I have made it clear that I am not talking about "applied algebra" here, so I don't quite see how introducing this distinction would have helped the question. On the other hand, the distinction between "structural" and "combinatorial" issues seems orthogonal to my question. The Burnside problem was proposed when everyone was already convinced that groups are something worth studying in their own right. I'm interested in the period before an algebraic concept reaches that stage.
Charles, I have to admit that I am having trouble distilling an answer to the question from the post and the above comment. I am not sure how to reconcile your two pieces of advice "follow the masters" and "Don't start off with self-peer-review". If I correctly read the latter as "don't try to predict whether the community will agree with you on your assessment of how interesting a topic is", then it seems to contradict the former.
Franz, maybe you could alert www-history.mcs.st-and.ac.uk to this. Kummer's biography there says "In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers. Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop."
Dear David, for the time being, I am more concerned about strategically prioritising different possible projects to advance my carreer than about the inner workings of the mathematical community. But the question addresses the broader issue of what we, the mathematicians, consider interesting to work on or to learn about. I hope (perhaps somewhat naively) that the two questions are very closely related. At any rate, my question is about the status quo, rather than about how people believe the world should work.
This is an encouraging story, thanks! I realise that this general question does not replace a specific discussion about the particular project I have in mind. But I feel that the issue is likely to repeat itself and applies to more people than just me. That's why I decided to phrase it in this generality.
I see what you are saying. I guess my question is then, what should happen first: should potential applications force the concept upon you or do you first introduce a concept and then let it find its place in an array of good stylines? Does the latter scenario work at all? For example, if people know that the more limited concept has its uses and you introduce the generalisation, they might actually start specifically looking for applications of the new thing. But will they? Or will the burden of proof of concept rest with the one introducing the generalisation?
That's a really nice example! I wonder, whether the algebraists would have caught on if the initial motivation hadn't been there. I know, these "what would have happened if"-questions are rarely sensible, but this example distills my question in a succinct way.
