How to prove it: $(-1)^{i\pi}$ is irrational?

Question

How to prove it: $(-1)^{i\pi}$ is irrational?

Here $i$ is the imaginary unit.

Thanks for the answers!!!

Answer 1

The answer is that it is probably still an open question whether that (multi-valued, depending on your definition) expression is irrational.

$-1=e^{i(2k+1)\pi}, k \in \mathbb{Z}$ by the broadest definition, so $(-1)^{(\pi i)} = e^{-(2k+1)\pi^2} $, so the values are all reciprocals of odd powers of $e^{\pi^2}$.

Now $e^{\pi}$ has long been known to be transcendental (and therefore irrational) by the Gelfond-Schneider theorem but the irrationality and transcendence (or otherwise) of $e^{\pi^2}$ is, to the best of my knowledge, an open problem.