Introduction to Modern Number Theory
Fundamental Problems, Ideas and Theories
Authors: Manin, Yu. I., Panchishkin, Alexei A.
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 About this book

"Introduction to Modern Number Theory" surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include nonAbelian generalizations of class field theory, recursive computability and Diophantine equations, zeta and Lfunctions.
This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects.
From the reviews of the 2nd edition:
"… For my part, I come to praise this fine volume. This book is a highly instructive read … the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly uptodate ..." (A. van der Poorten, Gazette, Australian Math. Soc. 34 (1), 2007)
 Reviews

From the reviews of the second edition:
"Here is a welcome update to Number theory I. Introduction to number theory by the same authors … . the book now brings the reader up to date with some of the latest results in the field. … The book is generally wellwritten and should be of interest to both the general, nonspecialist reader of Number Theory as well as established researchers who are seeking an overview of some of the latest developments in the field."
Philip Maynard, The Mathematical Gazette, Vol. 90 (519), 2006
[...] the first edition was a very good book; this edition is even better.
[...] Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject. [...]
Things get more interesting in Part II (by far the largest of the tree parts)[...] This part of the book covers such things as approaches through logic, algebraic number theory, arithmetic of algebraic varieties, zeta functions, and modular forms, followed by an extensive (50+ pages ) account of Wiles' proof of Fermat's Last Theorem. This is a valuable addition, new in this edition, and serves as a vivid example of the power of the "ideas and theories" that dominate this part of the book.
Also new and very interesting is Part III, entitled "Analogies and Visions,"
[...] The best surveys of mathematics are those written by deeply insightful mathematicians who are not afraid to infuse their ideas and insights into their outline of subject. This is what we have here, and the result is an essential book. I only wish the price were lower so that I could encourage my students buy themselves a copy. Maybe I'll do that anyway.
Fernado Q. Gouvêa, on 09/10/2005
"This book is a revised and updated version of the first English translation. … Overall, the book is very well written, and has an impressive reference list. It is an excellent resource for those who are looking for both deep and wide understanding of number theory." (Alexander A. Borisov, Mathematical Reviews, Issue 2006 j)
"This edition feels altogether different from the earlier one … . There is much new and more in this edition than in the 1995 edition: namely, one hundred and fifty extra pages. … For my part, I come to praise this fine volume. This book is a highly instructive read with the usual reminder that there lots of facts one does not know … . the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly uptodate … ." (Alf van der Poorten, Gazette of the Australian Mathematical Society, Vol. 34 (1), 2007)
 Table of contents (10 chapters)


Number Theory
Pages 961

Some Applications of Elementary Number Theory
Pages 6391

Induction and Recursion
Pages 95114

Arithmetic of algebraic numbers
Pages 115189

Arithmetic of algebraic varieties
Pages 191259

Table of contents (10 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Introduction to Modern Number Theory
 Book Subtitle
 Fundamental Problems, Ideas and Theories
 Authors

 Yu. I. Manin
 Alexei A. Panchishkin
 Series Title
 Encyclopaedia of Mathematical Sciences
 Series Volume
 49
 Copyright
 2005
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783540276920
 DOI
 10.1007/3540276920
 Hardcover ISBN
 9783540203643
 Softcover ISBN
 9783642057977
 Series ISSN
 09380396
 Edition Number
 2
 Number of Pages
 XVI, 514
 Additional Information
 Original Russian version of the first edition was published by VINITI, Moscow in 1990
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