Algebraic Number Theory

Algebraic Number Theory serves as a graduate-level text introducing the foundations and key concepts of algebraic number theory. The book progresses from basic definitions through increasingly complex topics including ideal theory, valuations, and local fields. Lang presents the material through a structured approach, building from elementary number theory and abstract algebra prerequisites to advanced concepts in class field theory. The text contains worked examples and exercises to reinforce the theoretical developments. The book covers major results like the Dirichlet unit theorem, ramification theory, and the fundamental theorems of algebraic number theory. Lang includes historical notes and references throughout to provide context for the mathematical developments. This text represents a systematic treatment of algebraic number theory that emphasizes rigor and abstraction while connecting various branches of mathematics. The presentation aims to develop both technical mastery and conceptual understanding of this core area of modern algebra.

Readers find Lang's Algebraic Number Theory to be dense and terse, with minimal motivation or examples. One PhD student noted it "reads like a reference manual rather than a textbook." Liked: - Comprehensive coverage of core topics - Clear, precise definitions and theorems - Good for reviewing concepts after learning them elsewhere - Strong on technical rigor Disliked: - Lack of motivation and context - Few examples or applications - Assumes significant background knowledge - Writing style described as "machine-like" and "cold" Several reviewers recommend using other texts like Neukirch or Marcus first. One mathematician wrote: "Only approach this after you already understand the basics from a gentler source." Ratings: Goodreads: 4.0/5 (43 ratings) Amazon: 3.7/5 (11 reviews) Mathematics Stack Exchange users frequently cite it as a solid reference text but not ideal for first exposure to the subject.

A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen Builds the foundations of algebraic number theory through elementary number theory and finite fields with a focus on explicit computations.

Introduction to Cyclotomic Fields by Lawrence C. Washington Examines cyclotomic fields, p-adic L-functions, and Iwasawa theory as natural extensions of classical algebraic number theory.

Algebraic Number Fields by Gerald J. Janusz Presents the arithmetic of algebraic number fields with emphasis on quadratic and cyclotomic fields, ideal theory, and Galois theory.

Algebraic Theory of Numbers by Pierre Samuel Connects classical number theory to modern abstract algebra through the study of number fields and their properties.

Number Fields by Daniel A. Marcus Develops the theory of algebraic numbers from first principles with detailed explanations of ramification theory and local fields.

🔢 First published in 1970, this text helped establish modern algebraic number theory notation and became a standard graduate-level reference work 📚 Serge Lang wrote over 45 mathematics textbooks during his career, earning him the nickname "The Machine" among colleagues for his prolific writing speed ⭐ The book introduces the concept of adeles and ideles - powerful tools in modern number theory that connect local and global properties of fields 🎓 Lang wrote this book while teaching at Columbia University, where he was known for his demanding teaching style and high expectations of students 🏅 The theories discussed in this book have practical applications in cryptography and are essential to understanding the proof of Fermat's Last Theorem, which wasn't solved until 1995