GelfondSchneider TheoremYou are currentlybrowsing as guest. Click here to log in 

In 1934, Gelfond and Schneider independently proved the following theorem, which now bears their names:
Suppose $a$ and $b$ are algebraic numbers. Then $a^b$ is transcendental unless a=0, a=1, or b is rational.
(Obviously, if any of those conditions hold then $a^b$ is in fact algebraic.)
This theorem implies that, for instance, $\sqrt{2}^\sqrt{2}$ and $e^\pi$ are transcendental. (The latter because otherwise $1=e^{i\pi}=(e^\pi)^i$ would be transcendental.)
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