Gelfond’s theorem

Let $\alpha$ and $\beta$ be algebraic over $\mathbb{Q}$, with $\beta$ irrational and $\alpha$ not equal to 0 or 1. Then ${\alpha }^{\beta }$ is transcendental over $\mathbb{Q}$.

This is perhaps the most useful result in determining whether a number is algebraic or transcendental.

The theorem is also known as the Gelfond-Schneider Theorem.

Title	Gelfond’s theorem
Canonical name	GelfondsTheorem
Date of creation	2013-03-22 13:24:31
Last modified on	2013-03-22 13:24:31
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	9
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 11J81
Synonym	Gelfond-Schneider Theorem
Related topic	LindemannWeierstrassTheorem
Related topic	Irrational