Hilbert's seventh problem

If $\alpha$ is an algebraic number with $\alpha \ne 0, 1$ and $\beta$ is an irrational algebraic number, is every value of the power $\alpha^\beta = \exp(\beta \log \alpha)$ necessarily transcendental? Equivalently, if $\alpha_1, \alpha_2$ are nonzero algebraic numbers whose logarithms have an irrational ratio $\log \alpha_1 / \log \alpha_2 \notin \mathbb{Q}$, must that ratio be transcendental?

Solved independently within months of each other in 1934 by Aleksandr Gelfond and Theodor Schneider (the Gelfond–Schneider theorem). Assuming for contradiction that $\alpha^\beta$ is algebraic, both proofs use Siegel's lemma to build a nonzero auxiliary function with integer coefficients (in Gelfond's version, a linear combination of $\exp((j + k\beta)z)$) that vanishes to high order on a grid of points, and then play an analytic upper bound from the maximum modulus principle against the algebraic lower bound $|\mathrm{Norm}(\gamma)| \ge 1$ for a nonzero algebraic integer $\gamma$.