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When solving an equation with irrational (or algebraic in the case) powers, are the roots likely to be transcendental or algebraic, or does it vary?

As an example, I was trying to figure out if $(x + 1)^{\sqrt{2}}=x^2 - 2 x + 2$ had an exact solution. I tried it myself and then let Wolfram Alpha work on it, only getting approximate solutions (0.31375... and 3. something)

I was curious if these solutions are algebraic, and thus the solution of some polynomial, or transcendental.

Also, are there any known techniques for solving this kind of thing?

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up vote 4 down vote accepted

Probably the solutions are almost always transcendental, but I don't know if this is provable in general with current technology. Known results along these lines include the Gelfond-Schneider and Lindemann-Weierstrass theorems.

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I believe Gelfond-Schneider proves they are not rational. – Ross Millikan Feb 18 '11 at 17:08
Gelfond-Schneider proves that the particular equation he's looking at has only transcendental roots, but it is not enough if there is more than one irrational power involved. Lindemann-Weierstrass might be able to handle some generalizations. – Qiaochu Yuan Feb 18 '11 at 17:16

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