Edit made on September 25, 2013 by ColinWright at 21:41:59
Deleted text in red
/
Inserted text in green
WW WM
HEADERS_END
In 1934, Gelfond and Schneider independently proved the following theorem, which now bears their names:
Suppose EQN:a and EQN:b are algebraic numbers. Then EQN:a^b is transcendental unless !/ a=0, a=1, !/ or /b/ is rational.
(Obviously, if any of those conditions hold then EQN:a^b is in fact algebraic.)
This theorem implies that, for instance, EQN:sqrt{2}^sqrt{2} EQN:\sqrt{2}^\sqrt{2} and EQN:e^\pi are transcendental.
(The latter because otherwise EQN:-1=e^{i\pi}=(e^\pi)^^i EQN:-1=e^{i\pi}=(e^\pi)^i would be transcendental.)
----
See also:
* http://www.google.co.uk/search?q=Gelfond-Schneider+Theorem
* http://en.wikipedia.org/wiki/Gelfond-Schneider_theorem
* http://mathworld.wolfram.com/GelfondsTheorem.html