Gelfond–Schneider theorem

In mathematics, the Gelfond–Schneider theorem is a result which establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and by Theodor Schneider. The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem.

[edit] Statement

If α and β are algebraic numbers (with α≠0 and

logα

any non-zero logarithm of α), and if β is not a rational number, then any value of

α β = exp{βlogα}

The values of $α$ and $β$ are not restricted to real numbers; all complex numbers are allowed.
In general, $α β = exp{βlogα}$ is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement.
An equivalent formulation of the theorem is the following: if $α$ and $γ$ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then $(logγ) / (logα)$ is either rational or transcendental.
If the restriction that $β$ be algebraic is removed, the statement does not remain true in general (choose $α = 3$ and $β = log2 / log3$ , which is transcendental, then $α β = 2$ is algebraic). A characterization of the values for α and β which yield a transcendental α^β is not known.

The transcendence of the following numbers follows immediately from the theorem:

The numbers $2^{\sqrt{2}}$ (the Gelfond–Schneider constant) and $\sqrt{2}^{\sqrt{2}}$ .
The number $e π$ (Gelfond's constant), as well as e^-π/2=iⁱ, since $e^{\pi} = e^{-i\,\log (-1)}$ is one of the values of $( - 1) - i$ .