Local fields number theory cambridge university press. Meanwhile the theory of p adic numbers has a plenty of applications and impacts in various mathematical. This is a common argument that is used in solving these types of equations. It covers valuations, hensels lemma, p adic analysis, and field extensions. Introduction to p adic numbers and valuation theory by bachman, george and a great selection of related books, art and collectibles available now at. In this chapter, we will explore divisibility, the building block of number theory. I am not a number theorist, but ive always had a distant fascination with p adic numbers. In algebra and number theory, one quite often deals with valuations instead of absolute values. Introduction to padic numbers an overview of ultrametric spaces and p adic numbers. The padic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. In mathematics, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. Slide 11 the p adics as a foundational tool hensel showed, in several papers and in his books, that one could give a simpler account of the theory of divisibility in algebraic number. Mathematics number theory and discrete mathematics.
Olympiad number theory through challenging problems. Seventh international conference on p adic in conclusion, i recommend this book to the set of people interested in p adic numbers, and its complement. Introduction to p adic analytic number theory download. The book treats the subject informally, making the text accessible to nonexperts. The padic numbers were invented and introduced to number theory by kurthensel 1 aroundtheyear1900motivatedbytheideaofbringingthe powerful tool of power series to this area of mathematics. The ams bookstore is open, but rapid changes related to the spread of covid19 may cause delays in delivery services for print products. The goal of this book is to offer such an opportunity, by way of a visit to the padic universe. William cottrell, p p adic strings and tachyon condensation, 2002. This book is an elementary introduction to p adic numbers. Borevich and shafarevich introduction to theory of numbers.
We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. This chapter will introduce many important concepts that will be used throughout the rest of the book. P adic analysis compared with real, by svetlana katok is a very gentle introduction to p adic numbers. Also doesnt do any local p adic theory, so you should pair it with gouveas excellent intro p adic book and you have great first course is algebraic number theory. 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. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a padic variable. As motivation for further study, the final chapter introduces iwasawa theory. Points of contact with other threads of the subject, including p divisible groups, p adic hodge theory, and rapoportzink spaces, are thoroughly explained.
This is the best introductionlevel mathematics textbook i have ever read, and i have read and own many. I have a list of neat math topics that i want to write about on my blog, and the p adic numbers are on that list. Tables of integrals of complexvalued functions of p. Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase.
Valenza, graduate text 186, springer 1999 fermats last theorem for amateurs, p. The p adic numbers were invented and introduced to number theory by. Here we follow some sections from the books 47, 96, 98. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. This elementary introduction offers a broad understanding of p adic numbers. This elementary introduction offers a broad understanding of padic numbers. Classfield theory, homological formulation, harmonic polynomial multiples of gaussians, fourier transform, fourier inversion on archimedean and p adic completions, commutative algebra. Marcuss number fields is a good intro book, but its not in latex, so it looks ugly. With over 100 exercises included, it will acquaint the nonexpert to. Introduction p adic numbers valuation theory abebooks. Some of his famous problems were on number theory, and have also been. The arizona winter school 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject.
Art of proofs pdf divisibility pdf olympiad number theory through challenging problems pdf third edition published december 18th, 2016 with the story behind the text. Algebraic numbers and algebraic integers, ideals, ramification theory, ideal class group and units, p adic numbers, valuations, p adic fields. Intermediate number theory pdf fourth edition last edited december 29th, 2017 first two chapters added. An introduction to algebraic number theory download book. Algebraic number theory studies the arithmetic of algebraic number. Thus, the search for p adic incarnations of the classical zeta and lfunctions is of relatively recent origin and has been a useful motif in the study of special values of various lfunctions and their arithmetic signi. Over the last century, padic numbers and p adic analysis have come to playa central role in modern number theory. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. Every theorem and definition is wellmotivated, and. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The p adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in advanced undergraduate or firstyear graduate courses.
Over the last century, padic numbers and padic analysis have come to playa central. The main objects that we study in this book are number. A resource for learning padic numbers mathematics stack exchange. This is a wellwritten introduction to the world of p adic numbers.
Im not entirely sure that mariano and gouvea are different people, i have never seen them in a room together. For those who dont know, defining the padic numbers involves. P adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Unlike many math books but like many teaching books, mathematical rigor mortis is. Such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis.
So i was happy to see an interesting article about them by andrew rich. This site is like a library, use search box in the widget to get ebook that you want. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Click download or read online button to get introduction to p adic analytic number theory book now.
Padic numbers, padic analysis and zetafunctions, 2nd edn. This text is suitable for an undergrad who has taken some analysis and topology. With over 100 exercises included, it will acquaint the nonexpert to the basic ideas of the theory. The basic library list committee recommends this book for. P adic numbers, p adic analysis and zetafunctions, 2nd edn. This is the first book to deal with applications of padic numbers in theoretical and mathematical physics. The reader is led into the rich structure of the fields qp and cp in a beautiful balance between analytic and algebraic aspects. Gouvea which seems nice, as mariano pointed out while i was typing. Schrijver, wiley 1999 fourier analysis on number fields, d.
Cassels textbook on local fields in the blue lms series. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. Lately padic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the nonarchimedean geometry of spacetime at small distances. Divisibility is an extremely fundamental concept in number. Algebraic number theory involves using techniques from mostly commutative algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects e. Most other books on the subject are written for more advanced students. Introduction to the theory of admissible representations of p adic reductive groups w. However, in section 9 we prove a new result, namely a p adic.
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. This textbook provides an introduction to p adic numbers suitable for upper undergraduate students. The aim of this talk is to give a survey on p adic numbers and p adic analysis. Another important contribution to the p adic theory was published in 1917byaleksandrmarkovi.
A course in p adic analysis, by alain robert is a more terse and advanced book on the subject. All three can be found in cassels book on local fields. This book is written for the student in mathematics. Branko dragovich, nonlocal dynamics of p adic strings, arxiv1011. A rather different direction is from the tradition of quadratic forms. The book teaches the use of open source software sage or gp, with lots of examples and exercises included throughout.
Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. Over the last century, padic numbers and padic analysis have come to playa central role in modern number theory. An 120 page introductory olympiad number theory book i wrote for anyone with a passion for number theory. In particular, two padic numbers are considered to be close when their difference is divisible by a high power of p. In recent decades, \ p \ adic geometry and \ p \ adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations.
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