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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).2012-03-13
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    In the link you can find Jim Brown's paper about the theorem. http://www.math.caltech.edu/~jimlb/abel.pdf2012-03-17
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    Qiaochu, sorry to misappropriate this question: At [this question](http://math.stackexchange.com/questions/118834/how-to-count-using-polynomials-when-we-have-subsets-that-partially-overlap), 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 :-)2012-03-19
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    @joriki: unfortunately I cannot read Chinese.2012-03-19

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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.2014-05-29