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## Transcendence III – Gelfond–Schneider–Lang

Here we prove among many other things the transcendence of and more generally the Gelfond–Schneider Theorem (see also Exercise 13.6) on αβ, which is allowed to be eβ log α for any interpretation of the logarithm. We illustrate the procedure with the Hermite–Lindemann Theorem of Chapter 13 on the transcendence of eα. Namely, if α and eα = β are both algebraic then they both lie in some number field K, and then so do the values , f () = βk (k = 0, 1, 2, . . .) of the functions z and f (z) = ez. Already we see hints of Chapters 9, 10 and 18.

Similarly for Gelfond–Schneider: if α and αβ = γ both lie ...

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