Gelfond-Schneider Constant is Transcendental

Theorem

$2^{\sqrt 2}$

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.

$2$ is algebraic.
$\sqrt 2$ is algebraic.
$\sqrt 2$ is irrational.

Hence the result.

$\blacksquare$

Historical Note

The question of the transcendental nature of the Gelfond-Schneider constant $2^{\sqrt 2}$ was raised in the context of the $7$th problem of the Hilbert $23$.

That the Gelfond-Schneider Constant is Transcendental was established in $1930$ by Rodion Osievich Kuzmin.

It was since determined to be a special case of the Gelfond-Schneider Theorem, established $\text {1934}$ – $\text {1935}$.