ABSTRACT
Around 1800, Ruffini sketched a proof, completed by Abel, that the general quintic equation is not solvable in radicals, by contrast to cubics and quartics whose solutions by radicals were found in the Italian renaissance, not to mention quadratic equations, understood in antiquity. Ruffini’s proof required classifying the possible forms of radicals. By contrast, Galois’ systematic development of the idea of automorphism group replaced the study of the expressions themselves with the study of their movements.
