Contents

• • Preface
  • Introduction
    • Mathematical background I assume you have
    • What is algebraic number theory?
      • Topics in this book
    • Some applications of algebraic number theory
  
  
• Classical Viewpoint
  • Finitely generated abelian groups
  • Commutative Algebra
    • Noetherian Rings and Modules
      • is Noetherian
  • Rings of Algebraic Integers
    • Rings of Algebraic Integers
    • Norms and Traces
  • Unique Factorization of Ideals
    • Dedekind Domains
  • Computing
    • Algorithms for Algebraic Number Theory
    • Using
      • Smith Normal Form
      • and Number Fields
      • Rings of integers
      • Ideals
  • Factoring Primes
    • Factoring Primes
      • A Method for Factoring that Often Works
      • A Method for Factoring that Always Works
      • Essential Discriminant Divisors
  • Chinese Remainder Theorem
    • The Chinese Remainder Theorem
  • Discrimannts, Norms, and Finiteness of the Class Group
    • Preliminary Remarks
    • Discriminants
    • Norms of Ideals
    • Finiteness of the Class Group via Geometry of Numbers
      • An Open Problem
  • Computing Class Groups
    • Remarks on Computing the Class Group
  • Dirichlet's Unit Theorem
    • The Group of Units
    • Finishing the proof of Dirichlet's Unit Theorem
    • Some Examples of Units in Number Fields
    • Preview
  • Decomposition and Inertia Groups
    • Galois Extensions
    • Decomposition of Primes
      • Quadratic Extensions
      • The Cube Roots of Two
  • Decomposition Groups and Galois Representations
    • The Decomposition Group
      • Galois groups of finite fields
      • The Exact Sequence
    • Frobenius Elements
    • Galois Representations and a Conjecture of Artin
  
  
• Adelic Viewpoint
  • Valuations
    • Valuations
    • Types of Valuations
    • Examples of Valuations
  • Topology and Completeness
    • Topology
    • Completeness
      • -adic Numbers
      • The Field of -adic Numbers
      • The Topology of (is Weird)
      • The Local-to-Global Principle of Hasse and Minkowski
    • Weak Approximation
  • Adic Numbers: The Finite Residue Field Case
    • Finite Residue Field Case
  • Normed Spaces and Tensor Products
    • Normed Spaces
    • Tensor Products
  • Extensions and Normalizations of Valuations
    • Extensions of Valuations
    • Extensions of Normalized Valuations
  • Global Fields and Adeles
    • Global Fields
    • Restricted Topological Products
    • The Adele Ring
    • Strong Approximation
  • Ideles and Ideals
    • The Idele Group
    • Ideals and Divisors
      • The Function Field Case
      • Jacobians of Curves
  • Exercises
  
  
• Bibliography
• Index