Galois Theory, 5th Edition

Chapter 26Further Directions

DOI: 10.1201/9781003213949-26

In this final chapter we give brief descriptions of other directions in which the ideas of Galois theory have been developed: the inverse Galois problem, differential Galois theory, and a brief introduction to p-adic Galois representations.

26.1 Inverse Galois Problem

A natural question, originally asked early in the 19th Century, is: Which finite groups can occur as the Galois group of a finite extension of the rationals $\mathbb{Q}$ ? Despite the simplicity of its statement, this question has not yet been answered completely. It is conjectured that the answer is ‘any finite group’, but this has neither been proved nor disproved. There are also generalisations; for example, let K be any field ...