Gelfond's Constant is Transcendental

Theorem
Gelfond's constant:
 * $e^\pi$

is transcendental.

Proof
From the Gelfond-Schneider Theorem:

If:
 * $\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \set {0, 1}$
 * $\beta$ is either irrational or not wholly real

then $\alpha^\beta$ is transcendental.

We have that:

As:
 * $i$ is algebraic
 * $-2 i$ is algebraic and not wholly real

the conditions of the Gelfond-Schneider Theorem are fulfilled.

Hence the result.