# Can a transcendental number raised to an algebraic power be algebraic?

March 2, 2013 6:33 AM Subscribe

Can a transcendental number such as pi, be raised to an irrational, but algebraic power resulting in an algebraic solution? Complex solutions would be acceptable. There might be a quick proof here, or there might not be. - Thanks for any help you can offer answering this! (And I promise that this isn't for a class or anything like that!)

Awesome, thank you! I feel silly now for not realizing. And that reference is great! Thanks so much!

posted by ch3cooh at 6:44 AM on March 2, 2013

posted by ch3cooh at 6:44 AM on March 2, 2013

In fact, just to emphasize the strength of the Gelfond-Scheider theorem: it says that if x is a real number with an irrational power which is algebraic, then not only

Transcendental number theory is not the most fashionable part of the subject right now, but it's really pretty cool.

posted by escabeche at 9:16 AM on March 2, 2013 [1 favorite]

*can*x be transcendental, but it*has*to be a transcendental number unless it is 0 or 1.Transcendental number theory is not the most fashionable part of the subject right now, but it's really pretty cool.

posted by escabeche at 9:16 AM on March 2, 2013 [1 favorite]

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Raise that number to the sqrt(2) power and you get 4.

The Gelfond-Schneider theorem will provide infinitely many examples of a similar flavor.

posted by escabeche at 6:37 AM on March 2, 2013 [12 favorites]