Gelfond's Theorem

Also called the Gelfond-Schneider Theorem. $a^b$ is Transcendental if

1. $a$ is Algebraic $\not= 0, 1$ and

2. $b$ is Algebraic and Irrational.

This provides the solution to the seventh of Hilbert's Problems.

See also Algebraic Number, Hilbert's Problems, Irrational Number, Transcendental Number

References

Baker, A. Transcendental Number Theory. London: Cambridge University Press, 1990.

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 107, 1996.