Gelfond–Schneider theorem explained

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.

History

It was originally proved independently in 1934 by Aleksandr Gelfond[1] and Theodor Schneider.

Statement

If a and b are complex algebraic numbers with a

\not\in\{0,1\}

and b not rational, then any value of ab is a transcendental number.

Comments

{\left(\sqrt{2}\sqrt{2

}\right)}^ = \sqrt^ = \sqrt^2 = 2.

Here, a is , which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if and, which is transcendental, then is algebraic. A characterization of the values for a and b which yield a transcendental ab is not known.

|a-1|p<1

and

|b-1|p<1,

then

(logpa)/(logpb)

is either rational or transcendental, where logp is the p-adic logarithm function.

Corollaries

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

2\sqrt{2

} and its square root

\sqrt{2}\sqrt{2

}.

e\pi=\left(ei\right)-i=(-1)-i=23.14069263\ldots

ii=\left(

i\pi
2
e

\right)i=

-\pi
2
e

=0.207879576\ldots

Applications

The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem.

See also

References

Further reading

External links

Notes and References

  1. Aleksandr Gelfond . Sur le septième Problème de Hilbert . Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na . VII . 4 . 623–634 . 1934 .