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Auxiliary Polynomials in Number Theory by David Masser

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19

Transcendence III – Gelfond–Schneider–Lang

Here we prove among many other things the transcendence of image 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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