Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics) [Hardcover]

Harold M. Edwards (Author)

Book Description

ISBN-10: 0387902309 | ISBN-13: 978-0387902302 | Publication Date: January 14, 2000 | Edition: 1st ed. 1977. Corr. printing

This introduction to algebraic number theory via the famous problem of "Fermats Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummers theory of "ideal" factorization. The more elementary topics, such as Eulers proof of the impossibilty of x+y=z, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummers theory to quadratic integers and relates this to Gauss'theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.

Editorial Reviews

From the Publisher

This book is a genetic introduciton to algebraic number theory which follows the development of the subject in the work of Fermat, Kummer and others, motivating new ideas and techniques by explaining the problems which led to their creation. The central problem is the one indicated in the title, but many other basic questions of algebraic number thoery are also treated. --This text refers to the Paperback edition.

Product Details

• Hardcover: 422 pages
• Publisher: Springer; 1st ed. 1977. Corr. printing edition (January 14, 2000)
• Language: English
• ISBN-10: 0387902309
• ISBN-13: 978-0387902302
• Product Dimensions: 9.2 x 6 x 1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,174,218 in Books (See Top 100 in Books)

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17 of 17 people found the following review helpful:

5.0 out of 5 stars Old school algebraic number theory with heavy Kummer bias, January 9, 2006

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics) (Hardcover)

Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

18 of 19 people found the following review helpful:

5.0 out of 5 stars great book, August 7, 2002

By A Customer

This review is from: Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics) (Hardcover)

This is a great book. If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want. it really has a lot of stuff in it. all other graduate books are theory without examples or motivation. this book is the exact opposite. the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

20 of 22 people found the following review helpful:

4.0 out of 5 stars Read this if you're seriously interested in math., July 24, 2001

By 

Victor A. Vyssotsky "mandvav" (Orleans, MA USA) - See all my reviews
(REAL NAME)   

This review is from: Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics) (Paperback)

There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book.