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The first correct proof of the insolvability of the quintic is due to Abel. But my question is who gave the proof of insolvability of the quintic using Galois theory? Does Abel know about Galois theory?

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    $\begingroup$ A quick skim of Wikipedia reveals, "in 1885, John Stuart Glashan, George Paxton Young and Carl Runge provided a proof using this theory [Galois theory]." $\endgroup$ – JamalS Feb 16 '15 at 7:58
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    $\begingroup$ What do you call "Galois theory"? I think Ruffini's proof used elements of permutation theory. Galois's proof certainly used permutations, and even permutation groups and concepts like normal subgroups. Or do you want a proof explicitly using the concept of an automorphism of a field? $\endgroup$ – Jack M Feb 16 '15 at 11:07

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