Softcover ISBN:  9780821847749 
Product Code:  AMSIP/27.S 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
Electronic ISBN:  9781470417420 
Product Code:  AMSIP/27.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $36.80 

Book DetailsAMS/IP Studies in Advanced MathematicsVolume: 27; 2002; 149 ppMSC: Primary 11;
This book is an elementary introduction to \(p\)adic analysis from the number theory perspective. With over 100 exercises, it will acquaint the nonexpert 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 nonexperts. It would make a nice independent text for a course geared toward advanced undergraduates and beginning graduate students.ReadershipAdvanced undergraduates, graduate students, and research mathematicians interested in number theory.

Table of Contents

Chapters

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


Reviews

The exposition of the book is clear and selfcontained. It contains numerous exercises and is wellsuited 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.
Mathematical Reviews


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This book is an elementary introduction to \(p\)adic analysis from the number theory perspective. With over 100 exercises, it will acquaint the nonexpert 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 nonexperts. It would make a nice independent text for a course geared toward advanced undergraduates and beginning graduate students.
Advanced undergraduates, graduate students, and research mathematicians interested in number theory.

Chapters

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

The exposition of the book is clear and selfcontained. It contains numerous exercises and is wellsuited 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.
Mathematical Reviews