Algebraic Number Theory - A Brief Introduction

Description

This book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic. The author presents the topic here by first offering a brief introduction to number theory and a review of the prerequisite material, then presents the basic theory of algebraic numbers. The treatment of the subject is classical but the newer approach discussed at the end provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. It leads naturally to the Weil conjecture and some delicate questions in algebraic geometry. Cover Half Title Series Page Title Page Copyright Page Contents Preface 1. Genesis: What Is Number Theory? 1.1. What Is Number Theory? 1.2. Methods of Proving Theorems in Number Theory 2. Review of the Prerequisite Material 2.1. Basic Concepts 2.2. Galois Extensions 2.3. Integral Domains 2.4. Factoring Rational Primes in Z[i] 3. Basic Concepts 3.1. Generalities 3.2. Algebraic Integers 3.3. Integral Bases 3.4. Quadratic Fields 3.5. Unique Factorization Property for Ideals 3.6. Ideal Class Group and Class Number 4. Arithmetic in Relative Extensions 4.1. Criterion for Ramification 4.2. Review of Commutative Algebra 4.3. Relative Discriminant for Rings 4.4. Direct Product of Rings 4.5. Nilradical 4.6. Reduced Rings 4.7. Discriminant and Ramification 5. Geometry of Numbers 5.1. Lattices in Rn 5.2. Minkowski's Lemma on Convex Bodies 5.3. Logarithmic Embedding 5.4. Units of a Quadratic Field 5.5. Estimates on the Discriminant 6. Analytic Methods 6.1. Preliminaries 6.2. The Regulator of a Number Field 6.3. Fundamental Domains 6.4. Zeta Functions 6.4.1. The Riemann Zeta Function 6.4.2. A Partial Zeta Function 6.4.3. The Dedekind Zeta Function 7. Arithmetic in Galois Extensions 7.1. Hilbert Theory 7.2. Higher Ramification Groups 7.3. The Frobenius Map 7.4. Ramification in Cyclic Extensions 7.5. The Artin Symbol 7.6. Quadratic Fields 7.7. The Artin Map 8. Cyclotomic Fields 8.1. Cyclotomic Fields 8.2. Arithmetic in Cyclotomic Fields 9. The Kronecker-Weber Theorem 9.1. Gauss Sums 9.2. Proof of the Kronecker-Weber Theorem 10. Passage to Algebraic Geometry 10.1. Valuations 10.2. Zeta Functions of Curves over Finite Fields 10.3. Riemann Hypothesis for Elliptic Curves over Finite 11. Epilogue: Fermat's Last Theorem 11.1. Fermat's Last Theorem 11.2. An Alternative Approach to Proving FLT Bibliography Index