Gel'fond-Schneider theorem

In mathematics, the Gel'fond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond:

α

is an algebraic number (with $\alpha\neq 0$ and $\alpha\neq 1$ ), and

β

α β

This statement implies that $2^{\sqrt{2}}$ (the Gelfond-Schneider constant) and $\sqrt{2}^{\sqrt{2}}$ (see nonconstructive proof) are transcendental numbers.

The Gelfond-Schneider theorem is a partial answer to Hilbert's seventh problem.

See also: Gel'fond-Schneider theorem, Algebraic number, Gelfond-Schneider constant, Hilbert's seventh problem, Irrational number, Mathematics, Nonconstructive proof, Transcendental number, Aleksandr Gelfond