From this question I thought: the algebraic numbers are closed under the operations $+,\div,\times,-,\sqrt[n]{\cdot}$ for arbitrary $n\in\Bbb N$.
However, as far as I know, they are not closed under exponentiation. Then my questions:
It is known if $\pi$ can be represented by $a^b$ for some algebraic numbers $a$ and $b$?
If the above question is positive, it is known if is possible to represent $\pi$ with the form $\left(\sqrt[n]{m}\right)^{\sqrt[p]{q}}$, for integers $n,m,p,q$?