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

March 2, 2013 6:33 AM

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!)
posted by ch3cooh to Education (3 answers total)

Best answer: 2^sqrt(2) is transcendental; this was a question of Hilbert, solved by Kuzmin in 1930.

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 [11 favorites]

Response by poster: 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

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 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]