TITLE: Is $e^e$ irrational?
QUESTION [15 upvotes]: I was surprised to find out that the following question is open:
Is $e^e$ transcedental?
According to Wikipedia, a positive answer to Schanuel's conjecture implies "yes" to the above question.
My questions:
1) Can we at least prove that $e^e$ is irrational? Or is this also open?
2) Given that $e^e$ is irrational, does it follow that $e^e$ is transcedental?
Added comment: For (2) I mean "Does the knowledge that $e^e$ is irrational help with the proof that $e^e$ is transcedental?"

REPLY [9 votes]: About (1), it is still unknown whether $e^e$ is irrational or not, according to Wikipedia.
https://en.wikipedia.org/wiki/Irrational_number#Open_questions
Even more interesting, according to Gelfond's Theorem, $a^b$ is transcendental (therefore irrational) if $a$ is algebraic (and $\not\in\{0,1\}$) and if $b$ is irrational and algebraic.
http://mathworld.wolfram.com/GelfondsTheorem.html
This theorem can be used to prove that $e^\pi$ is transcendental and therefore irrational.