The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I have recently become confused as to why this is the case.
The formula $z=\frac{-b+\sqrt{b^2-4ac}}{2a}$ expresses the solutions to the quadratic equation $az^2+bz+c = 0$ in terms of the inverse of an analytic function $z \mapsto z^2$. We have simply turned the problem of inverting one analytic function, $z \mapsto az^2+bz$, into the problem of inverting another analytic function, $z \mapsto z^2$. Therefore, all the power of the quadratic equation lies in how it solves any quadratic equation by inverting a single analytic function, $z \mapsto z^2$.
Similarly, Cardano's formula solves any cubic equation by inverting a two analytic functions, $z \mapsto z^2$ and $z \mapsto z^3$. Interestingly, you can also solve a cubic by inverting only one analytic function, for example $z \mapsto \sin z$.
And crucially, you can also do this for quintic equations, by inverting $z \mapsto z^k, k \leq 4$ and $z \mapsto z^5+z$.
One possible statement of the Abel–Ruffini theorem is that it is impossible to solve a general quintic equation by exclusively inverting functions of the form $z \mapsto z^k$ for $k \in \mathbb{N}$. But why would we only be interested in solutions that invert analytic functions of that form? In simplier terms, what's so special about radicals that makes solutions in terms of them so desirable? I can't see an argument that such inverses are intuitively straightforward: they often produce answers that purely formal (e.g. $\sqrt{2}$ doesn't have a simpler defintion than the positive inverse of the squaring function at $2$).
To me, it seems that the more natural question is, for $n \in \mathbb{N}$,
Is there a finite set of analytic functions such that the solutions to any degree $n$ polynomial may be expressed in terms of the inverse of these analytic functions?
I know very little about the status of this question (exept that it holds for some small values of $n$). Any information on what is known about this question would also be of interest.
Best Answer
I think that a large part of the difficulty we have in understanding why this result is considered important is that it is psychologically difficult to put oneself into the shoes of mathematicians of the past. There was a time not so many centuries ago when people didn't know how to solve cubic equations with radicals. Whether the quintic is solvable in radicals was once a difficult question. A problem that gains some notoriety for being difficult is usually going to be considered important when it is finally solved, regardless of whether it ends up occupying a central place in the "theory" that we end up constructing a posteriori. Fermat's Last Theorem is another good example of this.
Occasionally I will hear mathematicians say something to the effect of, "Fermat's Last Theorem isn't important; it's the math used to prove the theorem that is important." I don't entirely agree. There are two different kinds of importance that are being conflated. Something can be important because it occupies a central position in our theory. But an appealing and tantalizingly difficult problem is important because of its role in capturing our imagination and giving us something to sink our teeth into. I do not think we should disavow the importance of such problems just because they are solved. Many of our much-beloved theories would likely not exist if there hadn't been some interesting unanswered questions to motivate our research.
The quintic has the additional feature of being an "impossibility" result. Like non-Euclidean geometries and Gödel's incompleteness theorems, the solution made us realize that we had been making some unfounded assumptions about what the answer should look like. The psychological broadening of our horizons was a valuable byproduct that is sometimes overlooked.