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.

$\blacksquare$

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 $\text {1934}$ – $\text {1935}$.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental