Algebra and Geometry

Abstract

The purpose of this chapter is to sketch the classical background of abstract algebra. I shall introduce some famous mathematical problems, translate them into problems about equations, then try to indicate the difficulties that stand in the way of their solution. One reason for doing this is to demonstrate that algebra is central to mathematics. It is the natural setting for some of the oldest mathematical problems and, as we shall eventually see, it provides the keys to their solution. A second reason, complementary to the first, is to demonstrate the continuing relevance of classical mathematics. Even ancient mathematics is not dead or obsolete — it stays alive because its problems continue to stimulate the creation of new concepts and techniques. The most fertile problems in mathematics are over 2000 years old and still have not yielded up all their secrets.