Galois Theory, 5th Edition

Chapter 15Solution by Radicals

DOI: 10.1201/9781003213949-15

The historical aspects of the problem of solving polynomial equations by radicals have been 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 quintic polynomial equation whose Galois group is not soluble, namely the disarmingly straightforward-looking $t^5-6t+3=0$, which shows that the quintic equation cannot be solved by radicals.

Figure 23: Galois thought he had ...