Algebraic number theory and Fermat's last theorem

書誌事項

Algebraic number theory and Fermat's last theorem

Ian Stewart, David Tall

(A Chapman & Hall book)

CRC Press, c2016

4th ed

  • : hardback

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注記

Includes bibliographical references (p. 309-315) and index

内容説明・目次

内容説明

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: Mihailescu'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.

目次

Algebraic Methods Algebraic Background Rings and Fields Factorization of Polynomials Field Extensions Symmetric Polynomials Modules Free Abelian Groups Algebraic Numbers Algebraic Numbers Conjugates and Discriminants Algebraic Integers Integral Bases Norms and Traces Rings of Integers Quadratic and Cyclotomic Fields Quadratic Fields Cyclotomic Fields Factorization into Irreducibles Historical Background Trivial Factorizations Factorization into Irreducibles Examples of Non-Unique Factorization into Irreducibles Prime Factorization Euclidean Domains Euclidean Quadratic Fields Consequences of Unique Factorization The Ramanujan-Nagell Theorem Ideals Historical Background Prime Factorization of Ideals The Norm of an Ideal Nonunique Factorization in Cyclotomic Fields Geometric Methods Lattices Lattices The Quotient Torus Minkowski's Theorem Minkowski's Theorem The Two-Squares Theorem The Four-Squares Theorem Geometric Representation of Algebraic Numbers The Space Lst Class-Group and Class-Number The Class-Group An Existence Theorem Finiteness of the Class-Group How to Make an Ideal Principal Unique Factorization of Elements in an Extension Ring Number-Theoretic Applications Computational Methods Factorization of a Rational Prime Minkowski Constants Some Class-Number Calculations Table of Class-Numbers Kummer's Special Case of Fermat's Last Theorem Some History Elementary Considerations Kummer's Lemma Kummer's Theorem Regular Primes The Path to the Final Breakthrough The Wolfskehl Prize Other Directions Modular Functions and Elliptic Curves The Taniyama-Shimura-Weil Conjecture Frey's Elliptic Equation The Amateur Who Became a Model Professional Technical Hitch Flash of Inspiration Elliptic Curves Review of Conics Projective Space Rational Conics and the Pythagorean Equation Elliptic Curves The Tangent/Secant Process Group Structure on an Elliptic Curve Applications to Diophantine Equations Elliptic Functions Trigonometry Meets Diophantus Elliptic Functions Legendre and Weierstrass Modular Functions Wiles's Strategy and Recent Developments The Frey Elliptic Curve The Taniyama-Shimura-Weil Conjecture Sketch Proof of Fermat's Last Theorem Recent Developments Appendices Quadratic Residues Quadratic Equations in Zm The Units of Zm Quadratic Residues Dirichlet's Units Theorem Introduction Logarithmic Space Embedding the Unit Group in Logarithmic Space Dirichlet's Theorem Bibliography Index Exercises appear at the end of each chapter.

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