[FOM] Is Gelfond-Schneider constructive?

Alasdair Urquhart urquhart at cs.toronto.edu
Wed Mar 9 09:31:28 EST 2005

I had a look at the version of Gelfond's proof in Hua Loo Keng's
"Introduction to Number Theory."  It's less than 3 pages long
and fairly self-contained.  

The overall form of the proof is a reductio ad absurdum.
You suppose that alpha, beta and gamma = alpha^beta
are all in some algebraic number field.  You then introduce
an integral function R(x) with coefficients determined by a set
of homogeneous linear equations.  The proof then proceeds
through a sequence of explicit estimates, using Cauchy's integral
formula, ending with a refutable inequality.  

I haven't examined the proof in detail, but the reasoning 
appears to be constructive.

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