ABSTRACT
We give brief descriptions of three other directions in which the ideas of Galois theory have been developed. The inverse Galois problem asks: Which finite groups can occur as the Galois group of a finite extension of the rationals? The answer is thought to be ‘all of them’, but this has not been proved. Differential Galois Theory is a version of Galois theory that applies to differential equations. Finally, p-adic Galois representations occur in applications to algebraic number theory. At the current frontiers, the aim is to understand not just the Galois groups of specific finite extensions of the rationals, as in this book, but all of them at once. This involves topological rings: the familiar R and C, and the less familiar p-adic numbers.
