Galois Theory, 5th Edition

Chapter 14Solubility and Simplicity

DOI: 10.1201/9781003213949-14

In order to apply the Galois correspondence, in particular to solving equations by radicals, we need to have at our fingertips a number of group-theoretic concepts and theorems. We have already assumed familiarity with elementary group theory: subgroups, normal subgroups, quotient groups, conjugates, permutations (up to cycle decomposition): to these we now add the standard isomorphism theorems. The relevant theory, along with most of the material in this chapter, can be found in any basic textbook on group theory, for example Fraleigh (1989), Humphreys (1996), or Neumann, Stoy, and Thompson (1994).

We start by defining soluble groups and proving some basic properties. These ...