Lagrange on the prime degree equations solution

This report presents the results of the study of Lagrange's work "Reasoning about the solution of algebraic equations". Analyzing the methods of solving the third degree equations, Lagrange shows that all of them are based on the selection of the roots' functions of the given equation; for which the equation degree will be less than three. Lagrange notices that these roots' functions are multivalued. Their values depend on the position of the roots in the function. This fact leads Lagrange to permute the roots. From this permutation the roots' function acquires 3! values, however, Lagrange shows that the equation regarding the roots' function will have a second degree. Using the results obtained in the analysis the solutions of the third and fourth degree equation, Lagrange presents the analysis of the μ-degree full equation (with arbitrary μ) based on the Tschirnhaus' method. The report shows that in fact, Lagrange's research method is based on the group theory.Using the mechanism of this theory, Lagrange shows that for a given μ-degree equation (here we consider the case for μ is prime), the simplifying equation will be the equation depending of the roots' function of the given equation, whose degree will be μ-1, and the coefficients of this simplifying equation will satisfy to the equation of (μ-2)! degree. Lagrange, applying the results of this method to the fifth degree equation, shows that for its resolution all known methods cannot be used, because the degree of the simplifying equation increases.