The standard proof of the Abel-Ruffini theorem that people learn is based on Galois theory and the notion of a solvable group, but my understanding is that the original proof predates Galois theory. Where can I find a reasonably modern exposition of this original proof? Bonus points if it's online and free, of course.
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A friend of mine is reading Abel's Proof by Pesic. It seems to have what you're looking for and I think he is pretty happy with it. I can't personally vouch for it, though.