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P Adic Numbers

Author: Fernando Q. Gouvea
Publisher: Springer Science & Business Media
ISBN: 3662222787
Size: 19.21 MB
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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. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.

A Course In P Adic Analysis

Author: Alain M. Robert
Publisher: Springer Science & Business Media
ISBN: 1475732546
Size: 21.20 MB
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Discovered at the turn of the 20th century, p-adic numbers are frequently used by mathematicians and physicists. This text is a self-contained presentation of basic p-adic analysis with a focus on analytic topics. It offers many features rarely treated in introductory p-adic texts such as topological models of p-adic spaces inside Euclidian space, a special case of Hazewinkel’s functional equation lemma, and a treatment of analytic elements.

P Adic Analysis Compared With Real

Author: Svetlana Katok
Publisher: American Mathematical Soc.
ISBN: 082184220X
Size: 18.42 MB
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The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study.

Local Fields

Author: John William Scott Cassels
Publisher: Cambridge University Press
ISBN: 9780521315258
Size: 78.55 MB
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This book provides a fairly elementary and self-contained introduction to local fields.

P Adic Numbers P Adic Analysis And Zeta Functions

Author: Neal Koblitz
Publisher: Springer Science & Business Media
ISBN: 1461211123
Size: 47.80 MB
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The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, as well as numerous clarifications.

Mathematical Logic And Model Theory

Author: Alexander Prestel
Publisher: Springer Science & Business Media
ISBN: 1447121767
Size: 39.70 MB
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Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.

Problems In Algebraic Number Theory

Author: M. Ram Murty
Publisher: Springer Science & Business Media
ISBN: 0387269983
Size: 10.27 MB
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The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved

Introduction To Cyclotomic Fields

Author: Lawrence C. Washington
Publisher: Springer Science & Business Media
ISBN: 1461219345
Size: 73.80 MB
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This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.

Automorphic Forms

Author: Anton Deitmar
Publisher: Springer Science & Business Media
ISBN: 144714435X
Size: 22.89 MB
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Automorphic forms are an important complex analytic tool in number theory and modern arithmetic geometry. They played for example a vital role in Andrew Wiles's proof of Fermat's Last Theorem. This text provides a concise introduction to the world of automorphic forms using two approaches: the classic elementary theory and the modern point of view of adeles and representation theory. The reader will learn the important aims and results of the theory by focussing on its essential aspects and restricting it to the 'base field' of rational numbers. Students interested for example in arithmetic geometry or number theory will find that this book provides an optimal and easily accessible introduction into this topic.

Fermat S Last Theorem

Author: Harold M. Edwards
Publisher: Springer Science & Business Media
ISBN: 9780387950020
Size: 66.62 MB
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This introduction to algebraic number theory via "Fermat's Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummer theory of "ideal" factorization. In treats elementary topics, new concepts and techniques; and it details the application of Kummer theory to quadratic integers, relating it to Gauss theory of binary quadratic forms, an interesting connection not explored in any other book.