Worked solution: Gelfond–Schneider transcendence proof via auxiliary functions (1934)

In his 1900 address Mathematische Probleme Hilbert asked, as the seventh of twenty-three problems, whether $\alpha^\beta$ must be transcendental whenever $\alpha$ is algebraic with $\alpha\notin\{0,1\}$ and $\beta$ is algebraic irrational; here $\alpha^\beta$ means any fixed value of $e^{\beta\log\alpha}$ for a chosen branch of the logarithm, since different branches differ by powers of $e^{2\pi i\beta}$, which the theorem itself shows are transcendental. Hilbert cited $2^{\sqrt{2}}$ and $e^\pi=(-1)^{-i}$ as test cases and predicted the problem would outlast the Riemann Hypothesis.

Partial progress came quickly: Kuzmin (1930) handled real quadratic $\beta$ by a continued-fraction refinement of Gelfond's 1929 method, which had already proved $e^\pi$ transcendental. The general case fell in 1934 to Gelfond and, independently, Schneider; Schneider's variant was later recast by Siegel into the form used here, where one works with two entire functions rather than Gelfond's original single determinant-built function. The source cited in the file metadata is Gelfond's 1934 paper Sur le septième problème de Hilbert.

The argument is a quantified proof by contradiction, now the template for all of transcendence theory. Assume $\gamma=\alpha^\beta$ is algebraic and form the number field $K=\mathbb{Q}(\alpha,\beta,\gamma)$. One constructs a nonzero entire function $\Phi$, built from $e^z$ and $e^{\beta z}$, that vanishes to high order $L$ at the points $j\log\alpha$ where its values lie in $K$. A first nonzero derivative $\Phi^{(s)}(0)$ is then a nonzero element of $K$ that complex analysis (Jensen's formula) bounds above by $e^{-cL\log L}$ while algebraic number theory (the product formula over conjugates) bounds below by $e^{-CL}$; since $L\log L$ outruns $CL$, the two bounds collide, forcing $\gamma$ to be transcendental.