Algebraic Number Theory: A Graduate-Level Mathematics Textbook by A. Frohlich – Published by Cambridge University Press

Introduction

Algebraic number theory stands as one of the most elegant and profound branches of modern mathematics, bridging abstract algebra with the classical problems of integers and prime numbers. For graduate students and researchers in India seeking a rigorous yet accessible entry point into this field, Algebraic Number Theory by A. Frohlich offers a masterful blend of theoretical depth and computational practicality. Published by Cambridge University Press, this hardbound volume is an indispensable resource for anyone pursuing advanced studies in mathematics at Indian universities.

Book Overview

This book originates from graduate courses delivered at the University of Cambridge and the University of London, reflecting decades of pedagogical refinement. It provides a brisk yet thorough treatment of the foundational concepts of algebraic number theory, gradually introducing advanced ideas. The authors place strong emphasis on the systematic development of techniques for explicitly calculating basic invariants such as rings of integers, class groups, and units. Theory is consistently paired with concrete computations and applications, keeping the material grounded in classical number-theoretic problems. Special topics—including module theory of Dedekind domains, tame and wild ramifications, Gauss sums and periods, binary quadratic forms, and Brauer relations—are explored at a level usually found only in research monographs.

Key Highlights

• Rigorous Foundation: Builds algebraic number theory from first principles, assuming only a solid background in abstract algebra.
• Explicit Computations: Emphasizes hands-on calculation of rings of integers, class groups, and unit groups, making abstract concepts tangible.
• Special Topics: Includes advanced subjects like tame and wild ramification, Gauss sums, and Brauer relations rarely seen in introductory textbooks.
• Modern Algebraic Techniques: Combines clean, modern algebraic methods with substantial arithmetic content for a balanced approach.
• Proven Pedagogy: Based on actual graduate courses, ensuring logical flow and clarity suitable for self-study or classroom use.

Inside the Book

The text systematically covers the core structures of algebraic number theory: Dedekind domains, ideal theory, extensions of number fields, discriminant and different, and the geometry of numbers. Each chapter integrates theoretical developments with illustrative examples and exercises. The treatment of module theory over Dedekind domains is particularly detailed, providing a solid platform for understanding class groups. Later chapters delve into local fields, ramification theory (both tame and wild), and the arithmetic of quadratic forms. The inclusion of Gauss sums and periods offers a glimpse into analytic aspects, while Brauer relations connect the subject to representation theory. The book concludes with applications to classical problems, such as the representation of integers by binary quadratic forms.

Key Topics

• Rings of integers and Dedekind domains
• Ideal class groups and unit groups
• Discriminant, different, and ramification
• Module theory over Dedekind domains
• Tame and wild ramification in local fields
• Gauss sums, Gauss periods, and their applications
• Binary quadratic forms and class numbers
• Brauer relations and group actions

Reader Benefits

Readers will gain a deep, working knowledge of algebraic number theory that goes beyond mere memorisation. The explicit computational focus helps students develop the ability to calculate invariants for actual number fields, a skill essential for research. The inclusion of advanced topics prepares readers for current research literature, while the clear exposition makes even difficult concepts approachable. Indian students will find the algebraic perspective particularly valuable for competitive exams (like NET, GATE, and NBHM) and for pursuing doctoral work in number theory or related areas.

Learning Outcomes

• Master the structure of Dedekind domains and their ideal theory.
• Compute rings of integers, class groups, and unit groups for number fields.
• Understand ramification theory, including tame and wild cases.
• Apply Gauss sums and periods to classical number-theoretic problems.
• Analyse binary quadratic forms using class field theory concepts.
• Relate Brauer relations to the arithmetic of number fields.

Who Should Read

This book is ideal for graduate students in mathematics, especially those specialising in number theory, algebra, or arithmetic geometry. It also serves as a valuable reference for researchers in related fields such as cryptography, coding theory, or algebraic geometry. Indian students preparing for advanced exams or embarking on PhD research will find it an excellent companion. A strong background in abstract algebra (groups, rings, fields, Galois theory) is recommended.

About the Author

A. Frohlich was a distinguished mathematician and a leading figure in algebraic number theory, particularly known for his work on Galois module theory and the arithmetic of number fields. He held positions at the University of Cambridge and the University of London, where his teaching inspired generations of mathematicians. His research contributions remain influential, and this textbook reflects his gift for combining rigorous theory with practical computation.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers. With a history spanning over four centuries, it is renowned for producing authoritative textbooks and research monographs in mathematics and the sciences. This hardbound edition maintains the high production standards expected of a Cambridge publication, ensuring durability for years of study.

Conclusion

Algebraic Number Theory by A. Frohlich is a classic that has stood the test of time. For Indian students and researchers who want to master both the abstract foundations and the computational aspects of the subject, this book is an invaluable investment. Its blend of modern algebra, explicit calculations, and advanced topics makes it a unique resource that belongs on every serious mathematician's shelf. Order your copy today from Bookshops.in and deepen your understanding of one of mathematics' most beautiful disciplines.