Algebraic number theory and Fermat's last theorem Stewart, Ian

Table of Contents:


I Algebraic Methods


1. Algebraic Background

Rings and Fields
Factorization of Polynomials
Field Extensions
Symmetric Polynomials
Modules
Free Abelian Groups

2. Algebraic Numbers

Algebraic Numbers
Conjugates and Discriminants
Algebraic Integers
Integral Bases
Norms and Traces
Rings of Integers

3. Quadratic and Cyclotomic Fields

Quadratic Fields
Cyclotomic Fields

4. 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

5. Ideals

Historical Background
Prime Factorization of Ideals
The Norm of an Ideal
Nonunique Factorization in Cyclotomic Fields


II Geometric Methods

6. Lattices
Lattices
The Quotient Torus

7. Minkowski's Theorem

Minkowski's Theorem
The Two-Squares Theorem
The Four-Squares Theorem

8. Geometric Representation of Algebraic Numbers

The Space Lst

9. 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


III Number-Theoretic Applications


10. Computational Methods

Factorization of a Rational Prime
Minkowski Constants
Some Class-Number Calculations
Table of Class-Numbers

11. Kummer's Special Case of Fermat's Last Theorem

Some History
Elementary Considerations
Kummer's Lemma
Kummer's Theorem
Regular Primes

12. 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

13. 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

14. Elliptic Functions

Trigonometry Meets Diophantus
Elliptic Functions
Legendre and Weierstrass
Modular Functions

15. Wiles's Strategy and Recent Developments

The Frey Elliptic Curve
The Taniyama–Shimura–Weil Conjecture
Sketch Proof of Fermat's Last Theorem
Recent Developments

IV Appendices

A Quadratic Residues
Quadratic Equations in Zm
The Units of Zm
Quadratic Residues

B Dirichlet’s Units Theorem

Introduction
Logarithmic Space
Embedding the Unit Group in Logarithmic Space
Dirichlet's Theorem

Bibliography

Index

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.
Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(v14) 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.


(https://www.crcpress.com/Algebraic-Number-Theory-and-Fermats-Last-Theorem-Fourth-Edition/Stewart-Tall/9781498738392)