TRANSCENDENTAL NUMBER THEORY by Alan Baker – A Comprehensive Cambridge University Press Mathematics Textbook on Algebrai

Introduction

Transcendental Number Theory stands as a monumental work in modern mathematics, offering a deep and systematic exploration of numbers that cannot be expressed as roots of algebraic equations with rational coefficients. Authored by the legendary mathematician Alan Baker, this book has been a cornerstone for researchers, graduate students, and anyone passionate about number theory. Published by Cambridge University Press, this hardcover edition brings together decades of profound insights, making it an indispensable addition to any serious mathematical library in India and beyond.

Book Overview

First published in 1975, this classic text presents a rigorous yet accessible account of transcendental number theory. The book covers foundational concepts and advanced topics, including linear forms in logarithms of algebraic numbers, Schmidt's generalization of the Thue–Siegel–Roth theorem, Shidlovsky's work on Siegel's E-functions, and Sprindžuk's solution to the Mahler conjecture. This new edition features an insightful introduction by David Masser, which contextualizes Baker's achievement and explains the core arguments of his method. An updated afterword lists recent developments inspired by Baker's pioneering contributions.

Key Highlights

• Classic Authority: A definitive text by Fields Medalist Alan Baker, offering unmatched depth and clarity.
• Updated Edition: Includes a new introduction by David Masser and an afterword covering recent advances.
• Comprehensive Coverage: From basic principles to cutting-edge results, all in one volume.
• Rigorous Exposition: Every theorem is proved with meticulous detail, ideal for self-study or classroom use.
• Indian Edition: Published by Cambridge University Press, ensuring high-quality printing and binding for Indian readers.

Inside the Book

The book is structured to guide readers from foundational ideas to sophisticated theories. Early chapters introduce the concept of transcendental numbers and classical results like Liouville's theorem. Subsequent chapters delve into Baker's own groundbreaking work on linear forms in logarithms, which has profound applications in Diophantine equations. The later parts cover Schmidt's subspace theorem, Shidlovsky's theory of E-functions, and Sprindžuk's resolution of the Mahler conjecture. Each chapter is self-contained, with exercises and references that encourage further exploration.

Key Topics

• Linear Forms in Logarithms: Baker's celebrated theorem and its applications.
• Schmidt's Subspace Theorem: A powerful generalization of the Thue–Siegel–Roth theorem.
• Siegel's E-Functions: Shidlovsky's contributions to the theory of transcendental numbers.
• Mahler's Conjecture: Sprindžuk's solution and its implications.
• Classical Results: Liouville numbers, Hermite–Lindemann theorem, and Gelfond–Schneider theorem.

Reader Benefits

• Deep Mathematical Understanding: Gain a thorough grasp of one of the most elegant branches of number theory.
• Research Readiness: Equip yourself with tools to tackle open problems in transcendental number theory.
• Historical Insight: Appreciate the evolution of ideas from the 19th century to modern breakthroughs.
• Exam Preparation: Ideal for competitive exams like CSIR-NET, GATE, and PhD entrance tests in mathematics.
• Library Essential: A must-have reference for university libraries and personal collections.

Learning Outcomes

By studying this book, readers will be able to: understand the fundamental theorems of transcendental number theory; apply Baker's method to solve Diophantine equations; analyze the transcendence of specific numbers like e and π; evaluate proofs involving linear forms in logarithms; and connect transcendental theory with algebraic number theory and Diophantine geometry. The book also prepares students for advanced research in number theory and related fields.

Who Should Read

• Graduate Students: Pursuing master's or doctoral degrees in pure mathematics.
• Researchers: Mathematicians working in number theory, algebra, or related disciplines.
• Advanced Undergraduates: With a strong background in abstract algebra and real analysis.
• Teachers and Professors: Seeking a definitive resource for course design or reference.
• Self-Learners: Passionate mathematicians eager to explore transcendental numbers independently.

About the Author

Alan Baker (1939–2018) was a British mathematician who received the Fields Medal in 1970 for his groundbreaking work in number theory, particularly on transcendental numbers and linear forms in logarithms. He was a professor at the University of Cambridge and a fellow of Trinity College. His contributions continue to influence modern mathematics, and this book remains his magnum opus.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers. With a legacy spanning over four centuries, it is renowned for producing authoritative texts in mathematics, science, and humanities. This edition is part of their prestigious Cambridge Mathematical Library series, known for high editorial standards and durable hardcover binding.

Conclusion

Transcendental Number Theory by Alan Baker is not just a book—it is a gateway to understanding the profound mysteries of numbers. Whether you are a student beginning your journey in number theory or a seasoned researcher seeking a definitive reference, this classic work delivers unmatched depth and clarity. Order your copy today from Bookshops.in and add this masterpiece to your collection.