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In mathematics, the Gelfond-Schneider theorem is the following statement,
originally proved by Aleksandr Gelfond:
- If <math>\alpha<math> is an algebraic number (with <math>\alpha\neq 0<math> and <math>\alpha\neq 1<math>), and <math>\beta<math> is an irrational algebraic number, then <math>\alpha^{\beta}<math> is a transcendental number.
This statement implies that <math>2^{\sqrt{2}}<math> (the Gelfond-Schneider constant) and <math>\sqrt{2}^{\sqrt{2}}<math> (see nonconstructive proof) are transcendental numbers.
The Gelfond-Schneider theorem is a partial answer to Hilbert's seventh problem.
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