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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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Is galois-theory tag appropriate? (as you specifically don't want that). –  Aryabhata Mar 13 '12 at 22:52
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In the link you can find Jim Brown's paper about the theorem. math.caltech.edu/~jimlb/abel.pdf –  Mathlover Mar 17 '12 at 0:53
    
Qiaochu, sorry to misappropriate this question: At this question, which is pretty much incomprehensible (to me), I suggested to post the question in Chinese in the hope that someone might translate it. Unfortunately noone has; I thought perhaps you might be able to help or make a useful suggestion to the OP how to deal with this language problem. (I wouldn't ordinarily assume that you speak Chinese from your name, but I read something that said you were born in China :-) –  joriki Mar 19 '12 at 12:26
    
@joriki: unfortunately I cannot read Chinese. –  Qiaochu Yuan Mar 19 '12 at 14:57

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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.

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To be specific: Abel's Proof contains a full, commented translation of, well... Abel's proof, but not Ruffini's. –  Jack M May 29 at 9:02

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