Assuming integer co-efficients, does there exist a solution to a quintic polynomial that is a solution of radicals? I understand that there is no general formula to solve any quintic, but that doesn't necessarily mean that roots cannot be expressed in an algebraic formula. i.e one root may be $\sqrt[4]{2+\sqrt[3]{54}}$ and the other may be $\sqrt{33+\sqrt[5]{2}}$. That doesn't necessarily mean that they come from one single general formula.
Note: I am not familiar with the proof itself.
In fact, the usual proof of the Abel--Ruffini theorem shows, that for some particular choice of quintic (for example $x^5 - x - 1$) its roots cannot be written as expression involving arithmetic operations, radicals, and rational numbers (this of course implies the 'no general formula' version).