Polynomials in irrational powers and their roots

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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1 Answer

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