Solvability of Equations by Radicals

Abstract

Using the correspondence constructed in the preceding chapters, together with group-theoretic results, Galois obtained his famous criterion of solvability by radicals. “This material is so entirely new that new names and new characters are necessary to express it,” he wrote, adding later that the true value of his criterion is essentially theoretical, as it is often impossible to compute the Galois group of a given polynomial: “In a word, the computations are not practicable.” However, he adds, the applications generally lead to “equations all of whose properties are known beforehand,” so that the computations are possible, as in Chapters 9 and 10.