# Gelfond's Constant is Transcendental

## Theorem

$e^\pi$

## 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.

We have that:

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

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 $1934$ – $1935$.