Reference Request: Gelfond Schneider Theorem

Question

I want to learn the background for the Gelfond Schneider Theorem:

https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem

I want to learn the background needed for the proof of this theorem, as well as the proof itself. But I don't know what kind of textbook or class this would normally be covered in. Please recommend what I can study.

I want to be able to prove results on transcendence of specific numbers, which I know can be difficult in general. I also want to learn more about the subject in general, since my knowledge is very limited.

Thank you!!

Answer 1

The standard introductory books I know about are Irrational Numbers by Ivan Niven (1956; Gelfond-Schneider is the last chapter, Chapter X on pages 134-150) and Making Transcendence Transparent by Edward B. Burger and Rubert Tubbs (2004; Gelfond-Schneider is Chapter 5 on pages 113-146).

An expository paper on the proof is Gelfond's solution of Hilbert's seventh problem by Carl Einar Hille in American Mathematical Monthly 49 #10 (December 1942), pp. 654-661.

Less elementary is Transcendental Number Theory by Alan Baker. However, instead of Baker's book, perhaps better for you would be the 1972 Master of Science thesis Transcendental Numbers and a Theorem of A. Baker by Cameron Leigh Stewart.