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(1) Can anybody suggest a readable reference for Schneider's theorem that the number $$ \beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ is transcendental for $a, b \in \mathbb{Q}$ such that none of $a, b, a+b$ is an integer?

(2) Fix some integer $n \geq 3$ Is the degree of transcendence of the field generated over $\mathbb{Q}$ by $$ \left(\beta(\tfrac{i}{d}, \tfrac{j}{d})\right)_{i, j=1, \ldots, d-1} $$ known?

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Scheider's original paper is available online. The theorem you quote is proved at the end of Section 1, on Page 114. I don't know the answer to your second question, but my guess is "no".

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  • $\begingroup$ Thanks for the paper! Do you know other places where this is proved e.g. not in German? :) $\endgroup$ – schn93 Sep 19 '14 at 6:51

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