Algebraic Independence

We say that a set of m complex numbers is algebraically independent if there is no polynomial in m variables with rational coefficients whose value at those numbers is 0. Thus, to say that a number is transcendental boils down to the claim that a singleton set is algebraically independent. .

It is hard enough to prove that a specific number, say e or π, is transcendental. Showing that they are algebraically independent seems to be much harder. So hard, in fact, that no one has done it so far. We don’t even know, for example, that e + π is transcendental, though it surely must be.

Over the last decade or two Yuri Nesterenko has proved many results on algebraic independence, so a book by him on the subject is a good thing to have. This book is the result of a series of lectures given by the author at the Tata Institute in 1997. It is based on notes taken at the time by N. Saradha, which Nesterenko then worked up.

The book opens with a proof of the first important result in the field, known as the Lindemann-Weierstrass Theorem, which says that if α₁, α₂, …, α_n are distinct algebraic numbers, then e^α₁, e^α₂, …, e^α_n are linearly independent over the field of all algebraic numbers. It follows easily that both e and π are transcendental. Proving this takes 10 pages. Then we’re off to the races with more and more difficult results: the theory of E-functions, the Gelfond-Schneider Theorem, and finally Nesterenko’s own work.

The exposition is typical for transcribed lecture notes: spare and technical, with only a few general motivating comments. There are a few Slavicisms (e.g., “we first show that Lindemann-Weierstrass theorem is a consequence of Shidlovskii theorem”), but mostly the job of editing the text has been done very well.

For anyone interested in the area, this is definitely a book worth having.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.