Gelfond's Constant is Transcendental

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Gelfond's constant:


is transcendental.


From the Gelfond-Schneider Theorem:


$\alpha$ 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:

\(\displaystyle i^{-2 i}\) \(=\) \(\displaystyle \paren {e^{i \pi / 2} }^{- 2 i}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e^{-\pi i^2}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e^\pi\) $\quad$ $\quad$


$i$ is algebraic
$-2 i$ is algebraic and not wholly real

the conditions of the Gelfond-Schneider Theorem are fulfilled.

Hence the result.


Historical Note

The question of the transcendental nature of Gelfond's constant $e^\pi$ was raised in the context of the $7$th problem of the Hilbert $23$.

That Gelfond's Constant is Transcendental was initially established in $1929$ by Alexander Osipovich Gelfond.

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