Gelfond–Schneider theorem
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 and Theodor Schneider.Statement
If and are algebraic numbers with and irrational, then any value is a transcendental number.
Comments
The values of ' and ' are not restricted to real numbers; complex numbers are allowed.In general, is multivalued. This accounts for the phrase "any value of" in the theorem's statement.
An equivalent formulation of the theorem is the following: if ' and ' are nonzero algebraic numbers, and we take any non-zero logarithm of ', then is either rational or transcendental. This may be expressed as saying that if, are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The generalization of this statement to more general linear forms in logarithms of several algebraic numbers is in the domain of transcendental number theory.
If the restriction that a and b be algebraic is removed, the statement does not remain true in general. For example,
Here, is, which is transcendental rather than algebraic. Similarly, if and, which is transcendental, then is algebraic. A characterization of the values for ' and ' which yield a transcendental is not known.
Kurt Mahler proved the p-adic analogue of the theorem: if ' and ' are in, the completion of the algebraic closure of, and they are algebraic over ', and if and then is either rational or transcendental, where is the p''-adic logarithm function.
Corollaries
The transcendence of the following numbers follows immediately from the theorem:*