Basic Number Theory

Basic Number Theory is a graduate-level mathematics textbook published in 1964 by renowned mathematician Serge Lang. The book covers fundamental topics in algebraic number theory and presents core mathematical concepts with rigor and precision. The text progresses from elementary number theory through algebraic numbers, valuations, local fields, and global fields. Lang develops the material systematically, building from basic definitions to advanced theorems and applications in modern algebra. The book includes detailed proofs, exercises at varying difficulty levels, and connections to other areas of mathematics like topology and analysis. Its treatment of local-global principles and class field theory serves as a foundation for further study in algebraic number theory. The work exemplifies Lang's direct mathematical style and his view that complex ideas can be made accessible through careful exposition and logical development. Its influence on graduate mathematics education and research continues decades after its initial publication.

Readers report the book requires significant mathematical maturity, with graduate-level algebra and analysis as prerequisites. Multiple reviewers note it works better as a reference text than a self-study guide. Liked: - Comprehensive coverage of algebraic number theory - Rigorous proofs and detailed explanations - Strong focus on p-adic numbers and adeles - Clear treatment of class field theory Disliked: - Dense writing style makes concepts hard to grasp initially - Minimal motivating examples or intuitive explanations - Some proofs skip steps that readers must fill in - Dated notation compared to modern texts As one reviewer wrote: "Not for beginners. Lang assumes you're already comfortable with abstract algebra and can fill in the details yourself." Ratings: Goodreads: 4.0/5 (13 ratings) Amazon: 3.5/5 (4 ratings) Mathematics Stack Exchange: Frequently recommended for advanced graduate students, but not as a first text in the subject.

Algebraic Number Theory by J.W.S. Cassels and A. Frohlich This text provides comprehensive coverage of class field theory and local fields with a focus on their arithmetic properties.

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🔢 Written in 1964, this book grew from Lang's course notes at Columbia University and became one of the foundational texts in algebraic number theory. 🎓 The book was revolutionary in its approach by unifying local and global methods in number theory, a technique that wasn't commonly presented in textbooks at that time. ✍️ Serge Lang wrote over 100 mathematical texts during his career, but Basic Number Theory remains one of his most influential works, particularly for graduate students. 🌟 The book contains one of the clearest early presentations of adeles and ideles in number theory, concepts that are now considered essential in modern algebraic number theory. 🏆 The author, Serge Lang, was awarded the Cole Prize in 1960 for his work in number theory, just a few years before writing this book, adding to its authority in the field.