ESSENTIAL ALGEBRAIC NUMBER THEORY

This book provides a clear and accessible introduction to modern algebraic number theory, with a special emphasis on class field theory. Drawing from courses and lectures delivered in seven countries, the text balances simplicity with depth, making advanced topics approachable with minimal reliance on heavy algebraic or analytic machinery.

The book is structured into these distinct chapters; each tailored to a different stage of mathematical training:

The final chapter gathers a substantial collection of exercises, designed to test comprehension and guide further exploration. By offering the simplest known pathway to class field theory, this book fills a significant gap left by classic references published decades ago.

Contents:

• Preface
• About the Author
• Algebraic Number Fields:
  • Algebraic Prerequisites
  • Integrality
  • Dedekind Rings
  • p-adic Numbers
  • A Little about Class Field Theory
• Complete Discrete Valuation Fields:
  • Valuation Fields
  • Discrete Valuation Fields
  • Completion
  • Filtrations of Discrete Valuation Fie
  • Group of Principal Units as Topological ℤ_p-Module
  • Set of Multiplicative Representatives
  • Witt Ring
  • The Hensel Lemma and Henselian Fields
  • Extensions of Valuation Fields
  • Unramified and Ramified Extensions
  • Galois Extensions and Ramification Groups
  • Structure Theorems for Complete Discrete Valuation Fields
  • Cyclic Extensions of Prime Degree
  • Artin–Schreier Extensions
  • Hasse–Herbrand Function
  • Norm and Ramification Groups
  • Field of Norms
  • Local Fields with Finite Residue Fields
• Class Field Theory:
  • Main Results of Local Class Field Theory
  • Neukirch's Abstract Class Field Theory
  • Local Class Field Theory and Generalisations
  • Adeles of Global Fields
  • Zeta Functions and Zeta Integrals
  • Global Class Field Theory
• Exercises:
  • Algebraic Numbers Exercises
  • Local Fields Exercises
  • Class Field Theory and Zeta Functions Exercises
• Bibliography
• Index

 

Readership: Advanced undergraduates (third year and above) in mathematics. Master's students specializing in number theory or algebra. Doctoral students and postdoctoral researchers in algebraic number theory and related fields. Professors and academic researchers seeking a concise yet modern introduction to class field theory. Mathematicians in related research areas, such as algebra, algebraic geometry, algebraic topology, and mathematical physics, where number theory methods play a role.