My heart is inditing a good matter. Psalm 45.

After a general discussion of real-valued valuations of fields, attention will focus on the p-adic fields ℚp and their finite extensions. These provide the framework for much important and exciting research at the present day. They also give valuable insights at a humbler level, and not infrequently, provide remarkable easy and natural solutions to problems which apparently have no relation to p-adic fields and which otherwise can be resolved, if at all, only by deep and arduous methods.

The book supplies a self-contained introduction at the level of an MSc or beginning graduate student, though much will be of interest and accessible to the mathematical undergraduate or amateur. The aim is not to bring the reader to the frontiers of knowledge but, rather, to illustrate the versatility, power and naturalness of the approach. We therefore break the orderly exposition from time to time to make applications, some of which it is hoped that the reader will find striking. At the ends of the chapters are numerous exercises, ranging from the five-finger kind to substantial results of independent interest. The author will have failed if he does not persuade the reader that the p-adic numbers are every bit as natural and worthy of study as the reals and complexes.

In some of the applications and exercises we have assumed that the reader has access to a programmable calculator or home computer.