Gelfond–Schneider theorem (statement)

Theorem (Gelfond–Schneider, 1934). If $\alpha,\beta$ are algebraic numbers with $\alpha\notin\{0,1\}$ and $\beta\notin\mathbb{Q}$, then $\alpha^\beta$ is transcendental (and hence irrational).

Application. $\alpha=2$ (algebraic), $\beta=\sqrt2$ (algebraic, irrational). So $2^{\sqrt2}$ is transcendental.

Conclusion. $2^{\sqrt2}$ is transcendental. This solves Hilbert's seventh problem.