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Gelfond's theorem (Theorem)

Let $ \alpha$ and $ \beta$ be algebraic over $ \mathbb{Q}$, with $ \beta$ irrational and $ \alpha$ not equal to 0 or 1. Then $ \alpha^{\beta}$ is transcendental over $ \mathbb{Q}$.

This is perhaps the most useful result in determining whether a number is algebraic or transcendental.

The theorem is also known as the Gelfond-Schneider Theorem.



"Gelfond's theorem" is owned by mathcam. [ full author list (2) ]
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See Also: Lindemann-Weierstrass theorem, irrational

Other names:  Gelfond-Schneider Theorem
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Cross-references: number, transcendental, irrational, algebraic
There are 3 references to this entry.

This is version 6 of Gelfond's theorem, born on 2003-01-31, modified 2004-02-12.
Object id is 3952, canonical name is GelfondsTheorem.
Accessed 4533 times total.

Classification:
AMS MSC: 11J81 (Number theory :: Diophantine approximation, transcendental number theory :: Transcendence )
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Interpretation of $\alpha^\beta$ by archibal on 2004-02-15 23:35:29
Since $\beta$ is irrational, is this theorem intended to apply for any complex number $\alpha$, or simply for positive reals? If for any $\alpha$, should it then read "all possible values of $\alpha^\beta$"?
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