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.

tatement

:If α and β are algebraic numbers (with α≠0 and $log\; alpha$ any non-zero logarithm of α), and if β is not a rational number, then any value of is a transcendental number.

Comments

* The values of $alpha$ and are not restricted to real numbers; all complex numbers are allowed.
* In general, 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 $alpha$ and $gamma$ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then $(log\; gamma)/(log\; alpha)$ is either rational or transcendental.
* If the restriction that be algebraic is removed, the statement does not remain true in general (choose $alpha=3$ and , which is transcendental, then is algebraic). A characterization of the values for α and β which yield a transcendental α^β is not known.

Using the theorem

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 $e^\{pi\}$ (Gelfond's constant), as well as "e"^-π/2="i"ⁱ, since $e^\{pi\}\; =\; e^\{-i,log\; (-1)\}$ is one of the values of $(-1)^\{-i\}$.

ee also

* Lindemann-Weierstrass theorem
* Schanuel's conjecture; if proven it would imply both the Gelfond-Schneider theorem and the Lindemann-Weierstrass theorem

References

* "Irrational Numbers", by Ivan Niven; Mathematical Association of America; ISBN 0883850117, 1956
* [http://eom.springer.de/G/g130020.htm Gelfond-Schneider Method Entry in Springer's Encyclopedia of Mathematics]
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