Why do we make such big deal about the 'unsolvability' of the quintic? 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.
 A: For me there's a very simple and important answer, beyond what has been said already about the historical value of the work that was motivated by the problem.
And that answer is that the problem is natural and inherently interesting.
Why is any pure maths done? Why do we pay particular attention to some questions? Why, for example, do we care about the twin primes conjecture? It's not for any practical reason. If you mess around with numbers, it won't be long before you discover the concept of primes, and a while later you will notice twin primes and wonder how many of them there are. The question arises naturally as a product of observation and curiosity.
You may as well ask why we care so much about primes. "Sure", you say, "we can factor numbers into primes, but we can also factor numbers into things that aren't primes, so who cares?". The answer is that we find primes inherently interesting because of their simplicity (having the least possible number of divisors).
It's the same with roots and solving polynomial equations. If you play around with algebra, you'll naturally discover the concept of roots. You'll discover that low-degree polynomials can be solved using roots and nothing more, and you'll naturally ask whether this is true of all polynomials. Sure, you can also say that the solutions to $x^2 -2x -1 = 0$ are "whatever the solutions to $x^2 -2x -1 = 0$ are", but we have a special interest in solutions of the form $x^2 = a$ because of their simplicity, and so we find it interesting to learn that these are all that is required.
A: It is not a big deal anymore (edit: I mean in modern math, not for teaching undergraduate math courses) and has not been for a long time. It was important historically because of the math that came out of efforts to solve it, namely many important concepts in group theory (permutation groups, normal subgroups, quotient groups) and Galois theory.  Solvability of polynomials in radicals is often the culmination of a course on Galois theory but that result in no way reflects why Galois theory is important in contemporary math (which it is).
Galois theory in turn was very important for math even outside of algebra, e.g., the Galois correspondence inspired Lie to study Lie groups (to create a Galois theory for differential equations even if that is not how Lie theory is used today) and we can see parallels to it in the correspondence between covering spaces in topology and subgroups of the fundamental group.
Your emphasis on Abel-Ruffini and inverting analytic functions suggests you may have missed the point of what Galois did compared to Abel and Ruffini. The Abel-Ruffini theorem was about “generic polynomials” and was purely a negative result. Their work could not be applied to specific polynomials with rational coefficients, like $x^5-x-1$. Galois theory, in contrast, could be used on specific polynomials and works over fields that need not be inside the complex numbers.
By the way, Galois himself was pessimistic about the idea of actually applying his work to decide if a specific polynomial you did not already know something about is solvable by radicals. See pp. 225-231 of https://uberty.org/wp-content/uploads/2015/11/Peter_M._Neumann_The_Mathematical_Writings.pdf; the English translation is on pp. 227 and 229. The key part:

If you now give me an equation that you have chosen at will, and you wish to know whether or not it is solvable by radicals, I will have nothing to do other than to indicate to you the way to respond to your question, without wishing to charge either myself or anyone else with doing it. In a word, the calculations are impractical. [...] But most of the time in applications [...] one is led to equations all of whose properties one knows beforehand: properties by means of which it will always be easy to answer the question by the rules which we shall expound.

Even though taking roots is not enough to solve polynomial equations in algebra, inverting $z \mapsto z^k$ does suffice locally for maps between Riemann surfaces (created by Riemann 20 years after Galois) since every nonconstant holomorphic map between Riemann surfaces looks near each point like $z^k$ for a unique $k$ in suitable coordinates. So the phenomenon of ramification for such maps is based on the adequacy of power functions $z^k$ as a local model formula in single-variable complex analysis.
A: Perhaps it's not a big deal that's it's true; but it's a big deal that we can know and prove that it's true. I don't mean because the ability to prove that it's true implies the ability to do other, more useful things; I mean because it's not a kind of statement that seems like it could be proven true.
I distinctly recall learning as a young proto-mathematician that there was no way to double a cube or trisect an angle using straightedge and compass, no general solution to the quintic using radicals, and no closed-form integral of $\mathrm{e}^{-x^2}$, and I found it absolutely mind-boggling that these things could be true beyond "well, we haven't found a way to do it and we've looked really hard". This was before there were good resources for this kind of thing available on the internet, so to be honest, I still thought they might just be rumours that had been distorted in the retelling.
So as a high-school student it was hard to imagine that this is even a type of thing that can be known. A few years later I was an undergraduate at the University of Cambridge studying Galois Theory, and the lecture in which the insolubility of the quintic was proved (as a corollary of a much more general theorem) certainly felt like a milestone: look how far you have come.
A: Rephrasing in part some of the previous comments and answers, my take is that historically roots, like powers, were seen not as "functions" in the modern sense, but as natural "operations" extending the four standard ones. Then naturally mathematicians became curious about whether the resulting enlarged family of numbers (or "field" in modern language) would
A) consist always of solutions of polynomials;
B) contain all such solutions.
Solving the cubic was perhaps the major breakthrough in exploring B), then trying to solve the quintic became the inevitable stumbling block.
This remains a fascinating story because the historical path is after all not different from the natural development of many young adults' mathematical curiosity. As pointed out by others, we tend to forget it as our mathematical sophistication grows.
As an aside, I'd be curious to know when A) was settled historically. I'd be surprised if it hadn't been known at least by Lagrange or Euler, but then I'm not quite sure if rigorous notions of field extensions, degree, linear dependence were all in place then.
A: Mathematically, there is nothing special about radicals: solving algebraic equations by radicals is only one example of mathematical problems among others that are self-contained, comprehensive, complete and decidable in a given class of functions/expressions/numbers.
The question of solving algebraic equations by radicals comes from the historical question of solving algebraic equations by straightedge and compass. Today, this mathematical problem introduces Galois theory.
But there is also another branch: Topological Galois theory of Arnold and Khovanskii.
[Khovanskii 2021]:
"As was discovered by Camille Jordan the monodromy group of an algebraic function is isomorphic to the Galois group of the associated extension of the field of rational functions. Therefore the monodromy group is responsible for the representability of an algebraic function by radicals ... .
...
Theorem 12. If the monodromy group of an algebraic function is unsolvable then one can not represent it by a formula which involves meromorphic functions and elementary functions and uses integration, composition and meromorphic operations."
Let $\mathbb{E}$ be the explicit elementary numbers of [Chow 1999].
[Chow 1999]:
"Corollary 1. If Schanuel's conjecture is true, then the algebraic numbers in $\mathbb{E}$ are precisely the roots of polynomial equations with integer coeficients that are solvable in radicals."
That means: An algebraic equation that cannot be solved by radicals can neither be solved by explicit elementary numbers / in terms of explicit elementary functions nor by the other functions mentioned in Khovanskii's theorem 12 above.
$\ $
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Khovanskii 2014] Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014
[Khovanskii 2019] Khovanskii, A.: One dimensional topological Galois theory. 2019
[Khovanskii 2021] Topological Galois Theory - Slides 2021, University Toronto
A: To understand what is "so special about radicals that makes solutions of algebraic equations in terms of them so desirable" we should look at the problem not from the point of view of modern higher mathematics but from the point of view of mathematics as it was conceived possibly up to the fifties of the 20th century. Up to that time, there was not a clean distinction between pure and applied mathematics: mathematics was conceived as the "queen and servant to the other sciences". And in this context, asking

"Can I solve this algebraic equation in term of radicals?"

equals to ask

"Can I construct these intersections by using straightedge and compass (or just by using compass, after Georg Mohr and Lorenzo Mascheroni)?".

And knowing that there are some "elementary" constructions which I cannot do with such basic and ubiquitously used tools (not only in "R&D" offices but in every single mechanical workshop) means knowing that there are designs of some mechanical parts which are incomparably more difficult to achieve (in an exact way) respect to other, apparently similar, ones.
Thus, from the applied side of mathematics, it has been incomparably important to figure out this intrinsic limitation in the tools used, as this also in turn has forced the savants to develop new methods to design more and more complex mechanical parts: after all, we could also remember that Lorenzo Mascheroni, military engineer of Napoleone Bonaparte, developed his "compass" geometry in order to reduce the errors due to the use of a straightedge involved in making geometric constructions .
A: So many answers claiming that solving polynomials in radicals (read: finding their roots quickly) is not important! Here's my (and many others' who worked on graphics apps) train of thought at one time or another:
"Bézier curves make transitions very smooth because the first derivatives of adjacent sections coincide. Let's make the app even smoother with quintic curves so that the second derivatives coincide too!"
"Oh, wait, that would mean that I'd have to solve thousands of quintics per second, and I don't have a closed form solution for that! I would have to run thousands of interpolations rather than closed formula substitutions! Screw that; let's keep cubics."
A: The question of solvability by radicals led naturally to the study of solvable groups, and solvable groups play an important role in many parts of algebra and number theory, being in some sense the easiest groups after abelian groups. Just as one example, Wiles's proof of the modularity conjecture, and thus of Fermat's Last Theorem, relies on the fact that a certain small (non-abelian) matrix group is solvable, since that allows him to apply a theorem of Jerry Tunnell's on properties of Artin $L$-functions for Galois extensions of $\mathbb Q$ that have a solvable Galois group. So you could rephrase this by saying that Wiles' proof works because certain equations (roughly, that define the points of order 3 on the elliptic curve) are solvable by radicals. So this is an example of solvability by radicals arising in an unexpected place. And if, for example, Wiles needed to use the points of order 7, then (if I understand correctly) the proof would have fallen apart because we don't yet know enough about $L$-functions for fields that are not generated by radicals.
A: There are already some great philosophical reasons here, so let me give a very down-to-earth practical reason.
Suppose you have no electronic devices, but you have a great big book of logarithms. If you want to find the square root (or fifth root or whatever) of a number, you can do that in a few straightforward steps - look up the logarithm of the number, divide it by 2, and then do the reverse lookup to find which number has that as its logarithm.
What would be the quickest way to solve a cubic with this setup? Depending on the required precision, it could very well be to use the cubic formula! Of course there's Newton's method or other iterative methods, but the advantage of using the book of logarithms is you only need to do the calculation once and you already have the max precision.
For a general quintic there's no way to use the book of logarithms this way - you have to resort to iterative methods. I think that's noteworthy.
A: 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.
A: The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass.  Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would ever show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.
As for radicals, it is clear from Klein's work that solving x^n =a was considered "natural and explicit" because one can find the roots using logarithms.  (Set x = exp((1/n) log a).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas.  A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of +/-, *, division, and x -> x^(1/n).  Obviously something new has to be added,
and the radical operation is one of the simplest after the arithmetic operations.
In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.
In modern terms, one could argue that solving the equation x^n = a is distinguished and routine since Newton's method is highly reliable and rapid in this case.  Indeed for a>0, any real initial guess x>0 will converge and the number of digits of accuracy will double with each iteration.  For complex a, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero, indeed Hausdorff dimension < 2).
A: I like this question because I agree with its sentiment. Let me give an additional reason why the insolubility of the quintic is an overrated result in my opinion. I believe that we shouldn't even be saying that the general cubic is solvable in radicals. The reason being that we can't solve the general cubic if we restrict ourselves to the real numbers entirely, which I find an very reasonable restriction (see here). In my entire mathematical education, I've heard it mentioned only once that you need complex numbers to solve the general cubic, even though the roots may be real, which I find very surprising. Worse still, the supposed "solubility of the cubic" is widely advertised even in texts and presentations which are obviously aimed at audiences that are not necessarily familiar with complex numbers.
