Abel's impossibility theorem states that the roots of a general polynomial (of degree 5 or higher) cannot be written using arithmetic operations and radicals. Radicals are solutions of a specific polynomial basis and it's natural to wonder if roots of an arbitrary polynomial can be written in terms of the roots of a different polynomial basis.

Let $p_1, p_2, \ldots$ be a basis for the space of polynomials, where $p_i$ is of degree $i$. A "generalized radical" is a solution to $p_i = \alpha$ for some $i$ and constant $\alpha$. For example, when $p_i = x^i$, we get the standard notion of radicals.

It is then natural to think of another polynomial basis, such as falling factorials or some polynomial basis consisting of orthogonal polynomials. Is there a polynomial basis under which the generalized notion of radicals is powerful enough to express the roots of arbitrary polynomials?

Is there anything in the literature related to this problem?