Stewart I., Tall D. Algebraic Number Theory and Fermat’s Last Theorem

4th ed. — Boca Raton: CRC Press, 2016. — 314 p. — ISBN: 978-1-4987-3840-8.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.
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.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.Preface to the Third Edition
Preface to the Fourth Edition
Index of Notation
The Origins of Algebraic Number TheoryAlgebraic MethodsAlgebraic Background
Rings and Fields
Factorization of Polynomials
Field Extensions
Symmetric Polynomials
Modules
Free Abelian Groups
ExercisesAlgebraic Numbers
Algebraic Numbers
Conjugates and Discriminants
Algebraic Integers
Integral Bases
Norms and Traces
Rings of Integers
ExercisesQuadratic and Cyclotomic Fields
Quadratic Fields
Cyclotomic Fields
ExercisesFactorization 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
ExercisesIdeals
Historical Background
Prime Factorization of Ideals
The Norm of an Ideal
Non-Unique Factorization in Cyclotomic Fields
ExercisesGeometric MethodsLattices
Lattices
The Quotient Torus
ExercisesMinkowski’s Theorem
Minkowski’s Theorem
The Two-Squares Theorem
The Four-Squares Theorem
ExercisesGeometric Representation of Algebraic Numbers
The Space L^st
ExercisesClass-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
ExercisesNumber-Theoretic ApplicationsComputational Methods
Factorization of a Rational Prime
Minkowski Constants
Some Class-Number Calculations
Table of Class-Numbers
ExercisesKummer’s Special Case of Fermat’s Last Theorem
Some History
Elementary Considerations
Kummer’s Lemma
Kummer’s Theorem
Regular Primes
ExercisesThe 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
ExercisesElliptic 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
ExercisesElliptic Functions
Trigonometry Meets Diophantus
Elliptic Functions
Legendre and Weierstrass
Modular Functions
ExercisesWiles’s Strategy and Recent Developments
The Frey Elliptic Curve
The Taniyama–Shimura–Weil Conjecture
Sketch Proof of Fermat’s Last Theorem
Recent Developments
ExercisesAppendicesQuadratic Residues
Quadratic Equations in Z_m
The Units of Z_m
Quadratic Residues
ExercisesDirichlet’s Units TheoremLogarithmic Space
Embedding the Unit Group in Logarithmic Space
Dirichlet’s Theorem
ExercisesBibliography
IndexФайл: векторный PDF-файл (издат. макет) с закладками