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I am not sure if the following two types of transcendental names in the literature, so I will assign crudely a temporary moniker. We say a transcendental number $\alpha$ is of type I if for all non-zero algebraic numbers $\beta$, we have that $\alpha^\beta$ is transcendental. We say that $\alpha$ is of type II if there exists a non-zero algebraic number $\beta$ such that $\alpha^\beta$ is algebraic.

For example, it is known that $e$ is of type I, by the Lindemann-Weierstrass theorem. It is also known that there exist transcendental numbers of type II, established for example by the Gelfond-Schneider theorem. Here we can take $\alpha = 2^\sqrt{2}$ for example, and $\beta = \sqrt{2}$.

My question is has there any attempts to study transcendental numbers of type I or type II? To me it seems that transcendental numbers of type I are 'more' transcendental than those of type II. Is there any definition to make this intuition rigorous?

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A restatement of the definition of your "type II transcendental numbers": every real number of the form $x^y$ where $x$ is an algebraic number other than $0$ or $1$ and $y$ is an irrational algebraic number. In particular, of course, there are only countably many of them. – Robert Israel Feb 19 '12 at 21:29
Yes but that definition doesn't really help identify numbers such as $\alpha = e^\pi$, which is of type II, since $\alpha^i = -1$ is algebraic. There is no 'obvious' reason why $e^\pi$ should be of type II. – Stanley Yao Xiao Feb 20 '12 at 18:42

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