Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-called Bring radical, which is $a\mapsto f^{-1}(-a)$ for the monotomic polynomial $f(x)=x^5+x$.
[ Since my knowledge of this is a subset of what's on the Wikipedia page, it's not clear to me exactly how general the solution is; e.g. Wiki suggests that it gives extraneous solutions that must be carved out by numerical approximations, although perhaps this only causes problems for complex coefficients? ]
I am not particularly interested in formulas for roots of higher fixed-degree polynomials, but rather their feasibility in terms of roots of polynomials like $x^5+x$ that we deem to be "simpler" than general. My "real" question is: What is known about this topic, and/or what papers might an interested party look at to find out? To make this fit into the site's model, here are two more concrete questions that are interesting to me:
1. Classification: What I hope is meant by the general solution statement above is the following. $f(x)=x^5+x$ is a polynomial such that the roots of all degree-5 polynomials are some composition of: its inverse, $n^\text{th}$ roots, field operations, and the coefficients of the polynomial. Clearly $x^5+x$ is not unique in this regard, but I would be surprised if it is even "essentially unique" in any meaningful sense. To what extent can all polynomials $f$ with this property be described? (perhaps, when restricting to $\deg(f)\leq 5$?)
2. Complexity: Is there a "sextic Bring radical"? That is, I'm asking for a single polynomial (perhaps of very high degree?) that satisfies the general solution statement above, but with "degree-5" replaced by "degree-6". If so, is there a $d>6$ for which that no such "$d$-ic Bring radical" exists? If not, how few polynomials' inverses are required to solve a general sextic?
