Gelfond-Schneider Constant is Transcendental

Theorem
The Gelfond-Schneider constant:
 * $2^{\sqrt 2}$

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.

From Rational Number is Algebraic:
 * $2$ is algebraic.

From Square Root of 2 is Algebraic of Degree 2:
 * $\sqrt 2$ is algebraic.

From Square Root of 2 is Irrational:
 * $\sqrt 2$ is irrational.

Hence the result.