How does the Gelfond–Schneider theorem prove $e^\pi$ is transcendental?

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The Gelfond–Schneider theorem only gives criteria for $a^b$ being transcendental so long as both $a$ and $b$ are algebraic, but further, that $b$ is irrational.

I could understand arranging it to the form of $(e^{i\pi})^{-i}$, since $e^{i \pi}$ and $-i$ are both algebraic, but then how the hell is $i$ irrational? It's only one integer unit away from the complex origin.