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The Abel-Ruffini theorem provided the impetus for the development of the modern field of [[Galois]] theory, and indeed much of [[abstract algebra]].  Once some amount of Galois theory is understood, the impossibility of solving a general quintic turns out to be a consequence of some easy facts in [[group theory]]: in particular, the fact that the permutation group <math>S_5</math> is not "solvable".

The Abel-Ruffini theorem provided the impetus for the development of the modern field of [[Galois]] theory, and indeed much of [[abstract algebra]].  Once some amount of Galois theory is understood, the impossibility of solving a general quintic turns out to be a consequence of some easy facts in [[group theory]]: in particular, the fact that the permutation group <math>S_5</math> is not "solvable".