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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} and EQN:e^\pi are transcendental.

(The latter because otherwise EQN:-1=e^i\pi=(e^\pi)^i would be transcendental.)

(The latter because otherwise EQN:-1=e^{i\pi}=(e^\pi)^^i would be transcendental.)

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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