Gelfond's theorem, also called the Gelfond-Schneider theorem, states that  is transcendental if

1.  is algebraic  and

2.  is algebraic and irrational.

This provides a partial solution to the seventh of Hilbert's problems. It was proved independently by Gelfond (1934ab) and Schneider (1934ab).

This establishes the transcendence of Gelfond's constant  (since ) and the Gelfond-Schneider constant .

Gelfond's theorem is implied by Schanuel's conjecture (Chow 1999).

REFERENCES:

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

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 82-83, 2003.

Chow, T. Y. "What is a Closed-Form Number?" Amer. Math. Monthly 106, 440-448, 1999.

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.

Gelfond, A. O. "Sur le septième Problème de D. Hilbert." Comptes Rendus Acad. Sci. URSS Moscou 2, 1-6, 1934a.

Gelfond, A. O. "Sur le septième Problème de Hilbert." Bull. Acad. Sci. URSS Leningrade 7, 623-634, 1934b.

Schneider, T. "Transzendenzuntersuchungen periodischer Funktionen. I." J. reine angew. Math. 172, 65-69, 1934a.

Schneider, T. "Transzendenzuntersuchungen periodischer Funktionen. II." J. reine angew. Math. 172, 70-74, 1934b.