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Auxiliary Polynomials in Number Theory by David Masser

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22

Algebraic independence

We have seen plenty of this already, but only for functions over C. The property for numbers over Q is rather more difficult to prove. We quote some results relevant to the various chapters so far.

Regarding Chapter 2, we already mentioned that even the transcendence of the numbers in Theorem 2.1 is unknown.

Regarding Chapter 3, much is known. For example with Mahler’s image image one can prove the algebraic independence of f (α1), . . . , f (αm) for algebraic α1, . . . , αm with 0 < |α1|, . . . , |αm| < 1, provided a necessary and ...

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