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Dec 17, 2010 at 10:54 comment added Kevin H. Lin I recently (re)read the sections of Dummit-Foote on Galois theory. I think it was stated therein that Abel found the first proof of the insolvability of the quintic, but it was Galois who found the general method involving group theory for proving the solvability or insolvability of any polynomial. I don't recall if Dummit-Foote elaborates on Abel's method, but presumably it doesn't generalize like Galois's method does.
Dec 12, 2010 at 21:03 comment added Dylan Wilson Ah, I was searching for "Galois" and "Abel-Ruffini." Whoops...
Dec 12, 2010 at 11:14 comment added Gerry Myerson "Use of group theory to prove insolvability of 5th degree equation" is part of an earlier answer.
Dec 10, 2010 at 10:01 history answered Dylan Wilson CC BY-SA 2.5