Gelfond's Constant is Transcendental

Theorem

Gelfond's constant:

$e^\pi$

is transcendental.

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:

$\ds i^{-2 i}$

$=$

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

$\ds $

$=$

$\ds e^{-\pi i^2}$

$\ds $

$=$

$\ds e^\pi$

As:

$i$ is algebraic

$-2 i$ is algebraic and not wholly real

the conditions of the Gelfond-Schneider Theorem are fulfilled.

Hence the result.

$\blacksquare$

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 was initially established in $1929$ by Alexander Osipovich Gelfond.

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

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $23 \cdotp 140 \, 692 \, 632 \, 7792 \, 69 \, 005 \, 729 \, 086 \ldots$
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 9086 \ldots$