Comments on: Can a transcendental number raised to an algebraic power be algebraic?
http://ask.metafilter.com/236360/Can-a-transcendental-number-raised-to-an-algebraic-power-be-algebraic/
Comments on Ask MetaFilter post Can a transcendental number raised to an algebraic power be algebraic?Sat, 02 Mar 2013 06:33:30 -0800Sat, 02 Mar 2013 06:37:11 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: Can a transcendental number raised to an algebraic power be algebraic?
http://ask.metafilter.com/236360/Can-a-transcendental-number-raised-to-an-algebraic-power-be-algebraic
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!)post:ask.metafilter.com,2013:site.236360Sat, 02 Mar 2013 06:33:30 -0800ch3coohmathtranscendentalnumbertheorynumbersalgebraresolvedBy: escabeche
http://ask.metafilter.com/236360/Can-a-transcendental-number-raised-to-an-algebraic-power-be-algebraic#3425210
<a href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant">2^sqrt(2) is transcendental; this was a question of Hilbert, solved by Kuzmin in 1930.</a><br>
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Raise that number to the sqrt(2) power and you get 4.<br>
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The Gelfond-Schneider theorem will provide infinitely many examples of a similar flavor.comment:ask.metafilter.com,2013:site.236360-3425210Sat, 02 Mar 2013 06:37:11 -0800escabecheBy: ch3cooh
http://ask.metafilter.com/236360/Can-a-transcendental-number-raised-to-an-algebraic-power-be-algebraic#3425214
Awesome, thank you! I feel silly now for not realizing. And that reference is great! Thanks so much!comment:ask.metafilter.com,2013:site.236360-3425214Sat, 02 Mar 2013 06:44:18 -0800ch3coohBy: escabeche
http://ask.metafilter.com/236360/Can-a-transcendental-number-raised-to-an-algebraic-power-be-algebraic#3425262
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 <em>can</em> x be transcendental, but it <em>has</em> to be a transcendental number unless it is 0 or 1.<br>
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Transcendental number theory is not the most fashionable part of the subject right now, but it's really pretty cool.comment:ask.metafilter.com,2013:site.236360-3425262Sat, 02 Mar 2013 09:16:15 -0800escabeche