Algebraic Number Fields

Algebraic Number Fields serves as a graduate-level text covering the foundations and key concepts of algebraic number theory. The book progresses from basic number theory through to advanced topics including ideal theory and valuations. The content follows a systematic approach, building from fundamental definitions and theorems to applications in solving classical problems. Each chapter contains detailed proofs and exercises that reinforce the theoretical material. The text includes coverage of Dedekind domains, ramification theory, and the relationship between number fields and their rings of integers. Special attention is given to quadratic and cyclotomic fields as concrete examples. The book emphasizes rigorous mathematical development while maintaining connections to the historical origins and motivations of algebraic number theory. Its treatment balances abstract theory with computational methods that remain relevant to modern research in the field.

Readers describe this textbook as thorough and rigorous but challenging without prior exposure to abstract algebra and Galois theory. Liked: - Clear exposition of algebraic number theory fundamentals - Comprehensive exercises with solutions to odd problems - Strong coverage of ramification theory and class field theory - Effective progression from basic to advanced concepts Disliked: - Dense notation that can be hard to follow - Some proofs lack motivation/context - Prerequisites not clearly stated upfront - Few computational examples From a Goodreads review: "The book demands significant mathematical maturity. Not for beginners but excellent for those with the right background." Amazon reader: "Good reference text but difficult for self-study. Best used alongside lecture notes." Ratings: Goodreads: 4.0/5 (12 ratings) Amazon: 4.3/5 (8 ratings) Note: Limited review data available online as this is a specialized graduate mathematics text.

Algebraic Number Theory by J.W.S. Cassels and A. Frohlich. A compilation of foundational papers and developments in algebraic number theory that bridges classical and modern approaches.

Algebraic Theory of Numbers by Pierre Samuel. The text presents the core concepts of algebraic number theory with a focus on ideals, units, and quadratic and cyclotomic fields.

A First Course in Algebraic Number Theory by Algebraic Number Theory@@@ byHans Zassenhaus:::. This introduction connects elementary number theory with abstract algebra through the examination of number fields and their properties.

Introduction to Cyclotomic Fields by Lawrence C. Washington. The book develops the theory of cyclotomic fields and their applications to Fermat's Last Theorem and Iwasawa theory.

Number Fields by Daniel A. Marcus. The text constructs the foundations of algebraic number theory through Dedekind domains and ramification theory.

📚 The book was first published in 1973 and became a standard graduate-level text in algebraic number theory for several decades 🎓 Author Gerald J. Janusz taught at the University of Illinois at Urbana-Champaign and made significant contributions to the study of group rings and algebraic number theory 💫 Algebraic number fields, the book's subject, were first developed by mathematicians trying to solve Fermat's Last Theorem, though the theorem wasn't proven until 1995 🔢 The text includes detailed coverage of local fields and their completions, concepts that have important applications in modern cryptography and coding theory 📖 Unlike many contemporary texts, this book includes extensive historical notes and attribution of theorems to their original discoverers, providing valuable context for the development of the field