History of the impossibility theorem on the solution in radicals of polynomial equations

This report is devoted to the development of the proof of the possibility/impossibility of solving general equations in radicals. Lagrange has invented a method to analyze the function of roots of a given equation using substitutions. He has formally described the groups of substitutions. He has drawn attention to the question of whether the function of roots of a given equation belongs to a field of coefficients or to an extended field. This report shows the formation and the development of concepts such as group of substitutions and expansion of the set, as well as the appeal of this problem by later researchers such as Ruffini, Cauchy, Abel.