Introduction to $p$-adic Analytic Number Theory

This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises, it will acquaint the non-expert with the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of $p$-adic $L$-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the $p$-adic analog of the Riemann zeta function and $p$-adic analogs of Dirichlet's $L$-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for further study, the final chapter introduces Iwasawa theory. The book treats the subject informally, making the text accessible to non-experts. It would make a nice independent text for a course geared toward advanced undergraduates and beginning graduate students. Titles in this series are co-published with International Press, Cambridge, MA. Table of Contents: Historical introduction; Bernoulli numbers; $p$-adic numbers; Hensel's lemma; $p$-adic interpolation; $p$-adic $L$-functions; $p$-adic integration; Leopoldt's formula for $L_p(1,\chi)$; Newton polygons; An introduction to Iwasawa theory; Bibliography; Index. Review from Mathematical Reviews: The exposition of the book is clear and self-contained. It contains numerous exercises and is well-suited for use as a text for an advanced undergraduate or beginning graduate course on $p$-adic numbers and their applications...the author should be congratulated on a concise and readable account of $p$-adic methods, as they apply to the classical theory of cyclotomic fields...heartily recommended as the basis for an introductory course in this area. (AMSIP/27.S)