Given any equation, can we definitively determine in finite time whether or not this equation is solvable by non-numerical methods to give a symbolic answer?
It is trivial to show that there exist equations which is provable to have a symbolic answer (linear equations, quadratic equations, etc). Some equations may not have a symbolic answer: I do not know since my level of education is not high enough at the moment, I would admit.
I should make it clear, at this juncture, that no solutions exist is not the same as 'not having a symbolic answer', in that 'not having a symbolic answer' would mean we can prove the equation does not have a symbolic answer, but is or at least may be solvable by numerical methods to obtain an approximation. I am only looking for 'not having a symbolic answer' in this question.
Now given the above, is it possible, by algorithm or otherwise in theory, for us to determine, in finite time, for any given equation, whether or not it can be solved by symbolic, algebraic methods to give an exact, symbolic answer? Thank you very much.