Algebraic Number Theory

Algebraic Number Theory examines the foundations and core concepts of number fields, ideal theory, and valuations. The text progresses from basic definitions through advanced topics like class field theory and quadratic forms. Each chapter contains worked examples, exercises, and historical notes that connect the mathematical concepts to their origins. Applications to cryptography and computer science demonstrate the practical relevance of abstract number theoretic principles. The book maintains a balance between classical theory and modern developments in the field. Topics include Dirichlet's unit theorem, ideal class groups, local and global fields, and the fundamentals of algebraic integers. At its core, this text reveals the deep interplay between abstract algebra and number theory while highlighting the evolution of mathematical thought over centuries. The treatment connects elementary concepts to research-level mathematics in a structured progression.

Readers cite this as a solid introduction to algebraic number theory that bridges elementary and advanced concepts. The clear explanations and worked examples help build understanding. Liked: - Detailed proofs and step-by-step solutions - Historical context and motivations provided - Self-contained with review of prerequisites - Exercises with varying difficulty levels Disliked: - Some notation choices differ from standard conventions - Later chapters become more dense and challenging - A few typos and errors in problem solutions - Limited coverage of some advanced topics Ratings: Goodreads: 4.1/5 (14 ratings) Amazon: 4.0/5 (8 reviews) "The historical notes and motivating examples make abstract concepts more approachable" - Amazon review "Good first course text but prepare to work through details carefully" - Mathematics Stack Exchange user "Exercises helped reinforce understanding but solutions manual would be helpful" - Goodreads review

Algebraic Number Fields by Gerald J. Janusz This text builds from basic number theory to explore field extensions, ideal theory, and class groups with a focus on explicit computations.

A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen The book connects classical number theory with modern algebraic techniques through quadratic fields and cyclotomic extensions.

Number Fields by Daniel A. Marcus This work presents number field theory through concrete examples and emphasizes the computational aspects of algebraic numbers.

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

Algebraic Theory of Numbers by Pierre Samuel The book provides a concise treatment of algebraic number theory with a focus on ideal theory and unique factorization.

📚 Richard A. Mollin wrote over 250 research papers and authored 12 books during his career at the University of Calgary, where he was a Professor of Mathematics. 🔢 Algebraic number theory combines the power of abstract algebra with number theory, and its development was largely motivated by attempts to solve Fermat's Last Theorem. 🏛️ The foundations of algebraic number theory were laid by mathematicians Ernst Kummer, Richard Dedekind, and Leopold Kronecker in the 19th century. 💡 The book includes applications to cryptography, making it relevant to modern computer security and digital communication systems. 🎓 This text is considered advanced undergraduate or beginning graduate level, bridging the gap between basic number theory and more sophisticated algebraic concepts.