Hilbert's seventh problem

Solved, 1934Arithmetic and number theoryHilbert #7

Statement

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 $\gamma$ .

Posed by, 1900David Hilbert

Number fields and rings of integers

References

David Hilbert (1900). Mathematische Probleme
Aleksandr O. Gelfond (1934). Sur le septième problème de Hilbert
Theodor Schneider (1934). Transzendenzuntersuchungen periodischer Funktionen I. Transzendenz von Potenzen · DOI:10.1515/crll.1935.172.65
Alan Baker (1975). Transcendental Number Theory

History of the problem

1882Ferdinand von Lindemann proves that $e^\alpha$ is transcendental for every nonzero algebraic number $\alpha$ , establishing in particular the transcendence of $\pi$ via $e^{i\pi} = -1$ .Ferdinand von Lindemann
1900David Hilbert poses the seventh problem at the Paris ICM, naming $2^{\sqrt{2}}$ and $e^\pi = (-1)^{-i}$ as the flagship examples; in lectures he remarks that he expects to see the Riemann hypothesis solved before this problem.David Hilbert
1929Aleksandr Gelfond proves the case where $\beta = i\sqrt{d}$ is an imaginary quadratic irrational, immediately settling the transcendence of $e^\pi = (-1)^{-i}$ .
1930Rodion Kuzmin extends Gelfond's method to real quadratic irrationals $\beta = \sqrt{d}$ , proving that $2^{\sqrt{2}}$ is transcendental.
1934Aleksandr Gelfond and Theodor Schneider independently prove the full theorem for arbitrary algebraic $\alpha \ne 0, 1$ and arbitrary irrational algebraic $\beta$ .
1966Alan Baker generalizes Gelfond–Schneider to linear forms in arbitrarily many logarithms of algebraic numbers, with effective lower bounds that yield explicit bounds for Diophantine equations.