I've learned that polynomials of degree >= 5 (i.e. $x^5 - x -1$) are not necessarily solvable in the radicals, due to the Abel-Ruffini Theorem.
My question is: given that you can't solve a polynomial algebraically, what other (symbolic) methods exist that can solve it? That is, can I write down the roots using a "larger toolbox" than just algebraic operations? And, do these more powerful tools have analogous roles to group/field theory the way that radicals do?
For example - the group A5 has properties that prevent it from being solvable, where solvable means "has a derived series that terminates in the trivial subgroup". If I allow more operations than algebraic ones, then "solvable" would mean something different - maybe some new operation I bring in allows me get over a hurdle that I can't with radicals alone.
I have found some references to solving polynomials with ultraradicals and infinite series; I guess I'm specifically curious if there exist any symbolic methods that don't circumvent the whole Galois-group structure that we constructed to prove Abel-Ruffini in the first place, and in fact 'fit into' it in some interesting way.
To elaborate - I'd love to learn that there's a function f(x) that one can add to the basic toolbox of addition/multiplication/division/radicals that would make polynomials solvable. I imagine it's not exponentials or logarithms, of course - but surely there are many other functions in the world!