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 EditionProvides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper's proof that Z( is EuclideanPresents an important new result: MihA ilescu's proof of the Catalan conjecture of 1844Revises 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 TheoremImproves and updates the index, figures, bibliography, further reading list, and historical remarksWritten 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.