After a small search that I did I was unable to spot any answers here.What I am trying is to prove why the $2^{\sqrt2}$ is transcendental number. I know that this probably is a closed problem and probably many people have proved it already,but I want to reach the answer by myself after doing a research on this problem and going through several books and sources and test my knowledge and my power.So what I basically need is to give me guideline to solve the problem.If my question is too general,before you close it,please give me some information to rephrase,so it won't be so general.
2 Answers
$\begingroup$
$\endgroup$
1
By the Gelfond–Schneider theorem, $2^{\sqrt2}$ is transcendental.
$2^{\sqrt 2}$ is called the Gelfond–Schneider constant.

$\begingroup$ Thank you for your valuable help.Now at least I know where I can find the answer and what to look for to gain more knowledge. $\endgroup$– el10780Commented Sep 24, 2012 at 19:39
$\begingroup$
$\endgroup$
3
Don't worry, your question is wellstated. Some of these simply stated questions about transcendental numbers are still unknown! I don't know about this one. But here is another: Is $\pi^\pi$ a rational number? This one was unknown 20 years ago, and I expect it is still unknown.

$\begingroup$ Oh thank God.I am not experienced like you guys here.Thanks for your reply and thanks for giving me another problem to think about.(Although I do not think I will something like that.0 :) One think I can't understand though.Why they closed this question.The users who closed it claimed that it is offtopic.Offtopic compared to what?Did I misused the tagging? $\endgroup$– el10780Commented Sep 24, 2012 at 19:38

$\begingroup$ @el10780, this site is for researchlevel questions. For other questions, see math.stackexchange.com . $\endgroup$– lhfCommented Sep 25, 2012 at 2:25

$\begingroup$ Ok.Thank you for your respond and I apologize for my ignorance. $\endgroup$– el10780Commented Sep 25, 2012 at 5:39