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

In 1966 Alan Baker extended the Gelfond–Schneider theorem — which asserts the $\overline{\mathbb{Q}}$ -linear independence of two logarithms — to linear forms $\beta_1 \log \alpha_1 + \dots + \beta_n \log \alpha_n$ in arbitrarily many logarithms of algebraic numbers. Baker's quantitative lower bounds earned him the 1970 Fields Medal, yielded effective bounds on solutions to Thue, Mordell, and $S$ -unit equations, and provided a crucial ingredient in Mihăilescu's 2002 proof of Catalan's conjecture. Both Lindemann–Weierstrass and Baker's theorem would follow from Schanuel's conjecture, a sweeping open problem asserting that for any $z_1, \dots, z_n \in \mathbb{C}$ linearly independent over $\mathbb{Q}$ , the field $\mathbb{Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$ .

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.