It is not too hard to show that if Schanuel's conjecture is true, then the only algebraic numbers admitting a "closed-form expression" (as defined precisely in this paper) involving $e$, $\pi$, and other exponential-logarithmic constants are the ones solvable in radicals.

While reading Ken Ono's entertaining book *My Search for Ramanujan* recently, I was struck by the fact that some of Ramanujan's miraculous discoveries yield seemingly "transcendental expressions" for algebraic numbers. This leads to my question: Could one exploit special-function theory to construct an explicit closed-form expression for an algebraic number that is not solvable in radicals?

I expect the answer to be no, since I expect Schanuel's conjecture to be true. Still, even if that is the case, I wonder if there is any way to prove a precise theorem along these lines, that all closed-form expressions *constructed in a certain way* must be solvable in radicals if they are algebraic. Unfortunately, I am not familiar enough with special function theory to even tell if this question makes sense, but I was hoping some MO reader might be able to help.

**EDIT:** As an illustration, here's one of Ramanujan's results, reproduced from Douglas Hofstadter's book *Gödel, Escher, Bach*:

$${e^{-2\pi/\sqrt 5}\over\displaystyle \strut 1+{e^{-2\pi\sqrt 5}\over \displaystyle \strut 1+{e^{-4\pi\sqrt 5}\over \displaystyle \strut 1+{e^{-6\pi\sqrt 5}\over \displaystyle \strut {\ \atop 1+\cdots}}}}} = {{\sqrt 5 \over \displaystyle \strut 1+\root 5 \of {5^{3/4} \biggl({\sqrt 5 - 1 \over 2} \biggr)^{\! 5/2}\!\! -1}} -{\sqrt 5 + 1 \over 2}}$$ This doesn't answer my question directly because the expression on the left is an infinite continued fraction, which isn't a closed-form expression in my sense, but it does make me wonder whether there could be an identity of this sort where the "$\dots$" on the left-hand side can be replaced by a terminating expression.