
Chapter 15
Solution by Radicals
The historical aspects of the problem of solving polynomial equations by radicals
have been discussed in the introduction. Early in his career, Galois briefly thought
that he had solved the quintic equation by radicals, Figure 22. However, he found
a mistake when it was suggested that he should try some numerical examples. This
motivated his work on solubility by radicals.
The object of this chapter is to use the Galois correspondence to derive a con-
dition 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
quintic ...