So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots)
AS WELL AS a set of inverses for some polynomials which are not solvable using the aforementioned tools.
One such function can be given as:
$$y(x) = u| u^5 + u + x = 0$$
Is it possible given a set of extended tools to re-determine which polynomial equations have become solvable and which remain out of reach?
Furthermore....
What is the minimum set of inverse functions necessary to be able to make ALL polynomials solvable? I imagine this would be an infinite set so a better thing to ask is, what is a rule to generate a minimal set of polynomial inverse functions such that all polynomials are solvable given this set of inverse functions