In the second edition of this text, Koblitz presents a wide-ranging introduction to the theory of p-adic numbers and functions. In the first chapter he explains the details of building the p-adic rationals from the ordinary rational numbers. In the second, he delves into the p-adic version of the zeta function. In the third, he gives the algebraic closure of the p-adic rationals. In the final two chapters he returns to the study of p-adic functions, focusing on the zeta function again, but also the p-adic analogs of the exponential and gamma functions.
In addition to the theory presented, there are some really nice exercises that allow the reader to explore the material. Some of these problems are traditional and straightforward, but some are incredibly involved. So whatever your interest level is, there are problems here for you. And with the exercises, the book would make a good textbook for a graduate course, provided the students have a decent background in analysis and number theory.
Donald L. Vestal is an Associate Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu.
1: p-adic numbers. 2: p-adic interpolation of the Riemann zeta- function. 3: Building up _. 4: p-adic power series. 5: Rationality of the zeta-function of a set of equations over a finite field.