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?

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?

A related problem is whether there is a richer (but nontrivial) set of algebraic numbers than usual radicals that can capture the roots of arbitrary polynomials.

Is there anything in the literature related to these problems?