Updated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work.
New to the Fourth Edition
- Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean
- Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844
- Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem
- Improves and updates the index, figures, bibliography, further reading list, and historical remarks
Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.
Algebraic Methods: Algebraic Background. Algebraic Numbers. Quadratic and Cylclotomic Fields. Factorization into Irreducibles. Ideals. Geometric Methods: Lattices. Minkowski's Theorem. Geometric Representation of Algebraic Numbers. Class-Group and Class-Number. Number-Theoretic Applications: Computational Methods. Kummer's Special Case of Fermat's Last Theorem. The Path to the Final Breakthrough. Elliptic Curves. Elliptic Functions. Wiles's Strategy and Recent Developments. Appendices: Quadratic Residues. Dirichlet's Units Theorems.
"It is the discussion of [Fermat’s Last Theorem], I think, that sets this book apart from others — there are a number of other texts that introduce algebraic number theory, but I don’t know of any others that combine that material with the kind of detailed exposition of FLT that is found here...To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers."
—Dr. Mark Hunacek, MAA Reviews, June 2016
Praise for Previous Editions
"The book remains, as before, an extremely attractive introduction to algebraic number theory from the ideal-theoretic perspective."
—Andrew Bremner, Mathematical Reviews, February 2003