Algebraic Numbers and Algebraic Functions

Emil Artin's Algebraic Numbers and Algebraic Functions originated from his lectures at New York University during 1950-51. The text presents core concepts in algebraic number theory and function field theory from a unified perspective. The book progresses from foundational topics like valuations and completions to more advanced material on algebraic extensions. Each chapter builds systematically on previous concepts while maintaining clear connections between number fields and function fields. The treatment includes detailed discussions of ramification theory, different types of extensions, and the fundamental properties of algebraic function fields. Examples and exercises appear throughout to reinforce key ideas. This text serves as both a rigorous introduction to algebraic number theory and a demonstration of the deep parallels between number fields and function fields. The unified approach reveals underlying mathematical structures that connect these seemingly distinct areas.

Readers note this book preserves Artin's teaching style and lectures but find it requires significant mathematical maturity. Most comments indicate it works best as a supplementary text rather than a primary introduction. Liked: - Clear derivation of Dedekind rings and valuations - Concise treatment of classical algebraic number theory - Original insights from Artin's perspective - Quality of worked examples Disliked: - Dense presentation assumes strong background knowledge - Limited explanatory text between equations - Some passages feel incomplete or rushed - No exercises or practice problems Reviews are limited online since this is an advanced mathematics text. Goodreads: 4.67/5 (3 ratings) Amazon: 4.0/5 (2 ratings) A mathematics professor on MathOverflow wrote: "It maintains Artin's characteristic elegance and economy of presentation, but students may need additional references to fill in the gaps." No other substantive online reviews were found for this specialized text.

Basic Algebra by Anthony W. Knapp This text provides coverage of algebraic number theory and function fields with connections to Galois theory and complex analysis.

Introduction to Cyclotomic Fields by Lawrence C. Washington The book explores the theory of cyclotomic fields and their applications to number theory while building on the concepts of algebraic functions.

Algebraic Number Theory by Serge Lang The text develops the foundations of algebraic number theory with emphasis on class field theory and connections to algebraic functions.

Algebraic Function Fields and Codes by Henning Stichtenoth This work presents the theory of function fields over finite fields and their relationship to coding theory and algebraic geometry.

Class Field Theory by Nancy Childress The book builds the theory of algebraic number fields and their extensions through class field theory with concrete examples and computations.

🔢 Emil Artin developed this book from his lectures at New York University in 1950-51, but it wasn't published until 1967, several years after his death. 🎓 The book pioneered a unique approach to algebraic number theory by treating algebraic numbers and algebraic functions in parallel, highlighting their deep mathematical connections. ⚡ Artin fled Nazi Germany in 1937, bringing his mathematical expertise to America where he profoundly influenced the development of algebra at American universities. 📚 The text introduces the revolutionary concept of "Artin L-functions," which remain central to modern number theory and were later crucial in Andrew Wiles' proof of Fermat's Last Theorem. 🌟 Despite being based on graduate-level lectures, the book is known for its remarkably clear exposition, making complex concepts accessible through careful organization and well-chosen examples.