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
and
b not
rational, then any value of
ab is a
transcendental number.
Comments
- The values of a and b are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
- In general, is multivalued, where ln stands for the natural logarithm. This accounts for the phrase "any value of" in the theorem's statement.
- An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, 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 generalisation 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,
}\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.
and
then
is either rational or transcendental, where log
p is the
p-adic logarithm function.
Corollaries
The transcendence of the following numbers follows immediately from the theorem:
} and its square root
}.
e\pi=\left(ei\right)-i=(-1)-i=23.14069263\ldots
ii=\left(
\right)i=
=0.207879576\ldots
Applications
The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem.
See also
References
Further reading
-
- Book: Niven, Ivan . Irrational Numbers . registration . Ivan M. Niven . Mathematical Association of America . 1956 . 0-88385-011-7 .
External links
Notes and References
- 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 .