Gelfond's Constant is Transcendental

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Theorem

Gelfond's constant:

$e^\pi$

is transcendental.


Proof

From the Gelfond-Schneider Theorem:

If:

$\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \left\{{0, 1}\right\}$
$\beta$ is either irrational or not wholly real

then $\alpha^\beta$ is transcendental.


We have that:

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


As:

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

the conditions of the Gelfond-Schneider Theorem are fulfilled.

Hence the result.

$\blacksquare$


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 $1934$ – $1935$.


Sources