Algebraic number theory and Fermat s last theorem 3rd Edition by Ian Stewart, David Tall ISBN 978-1568811192 1568811195
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Product details:
ISBN 10: 1568811195
ISBN 13: 978-1568811192
Author: Ian Stewart, David Tall
First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat’s Last Theorem. Intended as a upper level textbook, it is also eminently suited as a text for self-study.
Table of contents:
I Algebraic Methods
1 Algebraic Background
1.1 Rings and Fields
1.2 Factorization of Polynomials.
1.3 Field Extensions
1.4 Symmetric Polynomials
1.5 Modules
1.6 Free Abelian Groups
1.7 Exercises
2 Algebraic Numbers
2.1 Algebraic Numbers
2.2 Conjugates and Discriminants
2.3 Algebraic Integers
2.4 Integral Bases
2.5 Norms and Traces
2.6 Rings of Integers
2.7 Exercises
3 Quadratic and Cyclotomic Fields
3.1 Quadratic Fields
3.2 Cyclotomic Fields
3.3 Exercises
4 Factorization into Irreducibles
4.1 Historical Background
4.2 Trivial Factorizations.
4.3 Factorization into Irreducibles
4.4 Examples of Non-Unique Factorization into Irreducibles
4.5 Prime Factorization
4.6 Euclidean Domains
4.7 Euclidean Quadratic Fields
4.8 Consequences of Unique Factorization
4.9 The Ramanujan-Nagell Theorem
4.10 Exercises
5 Ideals
5.1 Historical Background
5.2 Prime Factorization of Ideals
5.3 The Norm of an Ideal
5.4 Nonunique Factorization in Cyclotomic Fields
5.5 Exercises
II Geometric Methods
6 Lattices
6.1 Lattices
6.2 The Quotient Torus
6.3 Exercises
7 Minkowski’s Theorem
7.1 Minkowski’s Theorem
7.2 The Two-Squares Theorem
7.3 The Four-Squares Theorem
7.4 Exercises
8 Geometric Representation of Algebraic Numbers
8.1 The Space mathbf Lat
8.2 Exercises
9 Class-Group and Class-Number
9.1 The Class-Group
9.2 An Existence Theorem
9.3 Finiteness of the Class-Group
9.4 How to Make an Ideal Principal
9.5 Unique Factorization of Elements in an Extension Ring
9.6 Exercises
III Number-Theoretic Applications
10 Computational Methods
10.1 Factorization of a Rational Prime
10.2 Minkowski’s Constants
10.3 Some Class-Number Calculations
10.4 Tables
10.5 Exercises
11 Kummer’s Special Case of Fermat’s Last Theorem
11.1 Some History
11.2 Elementary Considerations
11.3 Kummer’s Lemma
11.4 Kummer’s Theorem
11.5 Regular Primes
11.6 Exercises
12 The Path to the Final Breakthrough
12.1 The Wolfskehl Prize
12.2 Other Directions
12.3 Modular Functions and Elliptic Curves
12.4 The Taniyama-Shimura-Weil Conjecture
12.5 Frey’s Elliptic Equation
12.6 The Amateur who Became a Model Professional
12.7 Technical Hitch
12.8 Flash of Inspiration
12.9 Exercises
13 Elliptic Curves
13.1 Review of Conics
13.2 Projective Space
13.3 Rational Conics and the Pythagorean Equation
13.4 Elliptic Curves
13.5 The Tangent/Secant Process
13.6 Group Structure on an Elliptic Curve
13.7 Applications to Diophantine Equations
13.8 Exercises
14 Elliptic Functions
14.1 Trigonometry Meets Diophantus
14.2 Elliptic Functions.
14.3 Legendre and Weierstrass
14.4 Modular Functions
14.5 The Frey Elliptic Curve
14.6 The Taniyama-Shimura-Weil Conjecture
14.7 Sketch Proof of Fermat’s Last Theorem
14.8 Recent Developments.
14.9 Exercises
IV Appendices
A Quadratic Residues
A.1 Quadratic Equations in Zm
A.2 The Units of Zm
A.3 Quadratic Residues
A.4 Exercises
B Dirichlet’s Units Theorem
B.1 Introduction.
B.2 Logarithmic Space
B.3 Embedding the Unit Group in Logarithmic Space
B.4 Dirichlet’s Theorem
B.5 Exercises
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