Gelfond-Schneider Theorem

Theorem
Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \left\{{0, 1}\right\}$.

If $\beta$ is irrational then any value of $\alpha^\beta$ is transcendental.

Proof
The Gelfond-Schneider Theorem and Some Related Results

Let $\alpha$ be an algebraic number such that $\alpha \ne 0$ and $\alpha \ne 1$.

Let $\beta$ be an algebraic number such that $\alpha^\beta$ is algebraic.

The result will follow if we can show that $\beta \in \Q$.

First we consider now the special case that $\alpha, \beta \in R$ and $\alpha > 0$.

It is enough to have $\ln \alpha \in \R$.

Observe that $\alpha^{s_1 + s_2 \beta}$ is an algebraic number for all integers $s_1$ and $s_2$.

To establish the result, it is enough to show that there are two distinct pairs of integers $(s_1, s_2)$ and $(s'_1, s'_2)$ for which:
 * $s_1 + s_2 \beta = s'_1 + s'_2 \beta$

We will choose $S$ sufficiently large and show such pairs exist with $0 \le s1, s2, s'_1, s'_2 < S$.

Lemma 1
This was Problem 7 in the Hilbert 23.