Essential Algebraic Number Theory (Series on Number Theory and Its Applications)

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: Chapter 1 introduces algebraic number fields, progressing at a steady pace with numerous examples — ideal for advanced undergraduates; Chapter 2 explores complete discrete valuation fields (local fields), a vital area of local algebraic number theory often underrepresented in standard textbooks but crucial in modern research; Chapter 3 develops abstract class field theory and its applications to both local and global fields, using Neukirch's axiomatic approach to derive fundamental theorems. Extensions to recent developments and generalizations are also discussed. 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. Read more