Gelfond–Schneider theorem

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond[1] and Theodor Schneider. The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem.


If a and b are algebraic numbers with a  0,1 and b irrational, then any value of ab is a transcendental number.


Here, a is 22, which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if a = 3 and b = (log 2)/(log 3), which is transcendental, then ab = 2 is algebraic. A characterization of the values for a and b, which yield a transcendental ab, is not known.


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

See also


  1. Aleksandr Gelfond (1934). "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.

External links

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