Transcendental Number Theory

Cambridge Mathematical Library

by Alan Baker, David Masser (Foreword)

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is, the theory of those numbers that cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory, which continues to see rapid progress today. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue–Siegel–Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindžuk's solution to the Mahler conjecture. This edition includes an introduction written by David Masser describing Baker's achievement, surveying the content of each chapter and explaining the main argument of Baker's method in broad strokes. A new afterword lists recent developments related to Baker's work.

Genres: Mathematics, Algebra

Kindle Edition

First published April 25, 1975

About the author

Alan Baker was an English mathematician, known for his work on effective methods in Number theory, in particular those arising from transcendence theory. He was awarded the Fields Medal in 1970, at age 31. His academic career started as a student of Harold Davenport, at University College London and later at Cambridge. He was a visiting scholar at the Institute for Advanced Study in the fall of 1970. He is a fellow of Trinity College, Cambridge. His interests were in number theory, transcendence, logarithmic form, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mathematical Society.

Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.

Reviews

[Simon Mcleish · Rating 3 out of 5]

Originally published on my blog here in June 1998.

This book has been the standard survey of the theory of transcendental numbers for some time now - the first edition was published in the mid-seventies. The author is a prominent researcher in the field, and several chapters draw heavily on his own work.

My own mathematics training is in areas other than number theory, and I found large parts of the book difficult to follow. I have a copy of the basic number theory textbook written by Baker, A Concise Introduction To The Theory Of Numbers, and I would have expected some discussion of concepts used in Transcendental Number Theory but absent from the more basic book.

The other problem is with Baker's writing style, and it is shared with An Introduction to Number Theory. His mathematics is extremely condensed in style, and it is often difficult to work out what is going on. In a survey this is less of a problem than in a textbook, but I still felt I would have found it easier to follow if I were attending a course of lectures based around the book at the same time as reading it.