Is $e^{\pi \alpha}$ known to be transcendent for all real algebraic $\alpha$?

Question

The MathWorld article Transcendental Number contains a reference to Yu. V. Nesterenko proof of transcendence of $e^{\pi \sqrt{2}}$. Is there a more general result about transcendence of $e^{\pi \alpha}$ for all real algebraic $\alpha$?

I observed that WolframAlpha returns an affirmative answers for questions like "Is exp(pi * tan(pi/ 17)) a transcendental number?"

Answer 1

Of course, I missed an obvious counterexample $\alpha=0$. But it seems this is the only counterexample, i.e. $e^{\pi \alpha}$ is a transcendental number for all non-zero real algebraic $\alpha$. It follows from the Gelfond–Schneider theorem:

If $a$ and $b$ are algebraic numbers where $a \notin \{0,\,1\}$ and $b \notin \mathbb{Q}$, then any value of $a^b$ is a transcendental number.

Just let $a=i$ and $b = -2 i \alpha$. Now we have $b \notin \mathbb{Q}$ because $\Im(b) \ne0 $.