TITLE: Product of two transcendental numbers is transcendental
QUESTION [5 upvotes]: Let $\alpha,\beta$ be transcendental numbers. Which of the followings are true?
1)$\alpha\beta\ \text{ is transcendental}$.
2)$\mathbb{Q}(\alpha)\ \text{is isomorphic to }\mathbb{Q}(\beta)$
3)$\alpha^\beta\ \text{is transcendental }$
4)$\alpha^2\ \text{is transcendental}$
I know option 4 is true. And I feel option 1 is also true, but I don't know the exact  reason. While I have no idea  for the remaining two.

REPLY [4 votes]: Here is a proof that point 3 is false that doesn't use any difficult theorems.
As $\gamma$ ranges through all transcendental numbers, the number $2^\gamma$ takes on uncountably many values. Since only countably many of these values can be algebraic, there must exist a transcendental $\gamma$ for which $2^{\gamma}$ is also transcendental. 
Now let $\alpha = 2^{\gamma}$ and $\beta = 1/\gamma$.