ABSTRACT
The historical aspects of the problem of solving polynomial equations by radicals were discussed in the introduction. The object of this chapter is to use the Galois correspondence to derive a condition that must be satisfied by any polynomial equation that is soluble by radicals, namely: the associated Galois group must be a soluble group. We then construct a specific quintic polynomial equation whose Galois group is not soluble, which shows that the quintic equation cannot be solved by radicals.
