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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!

Alex Kritchevsky
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    See this question. There are symbolic methods (algorithms) to find the roots approximately, e.g. $x^5-x-1$ has only one real root, namely $x=1.16730397826$. – Dietrich Burde May 25 '16 at 19:39
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    what about newton/raphson method. That should work to find zeros. Give accurate results to any degree. Only requirement that the polynomial have defined derivative. – marshal craft May 25 '16 at 19:44
  • I'm specifically interested in methods that have "theoretical" analogs, the same way that regular factorization has an analog when you look at polynomials under the lens of group theory. Numerical approximations and infinite series solutions are useful, but I think they're circumventing what I'm interested in. – Alex Kritchevsky May 25 '16 at 22:42
  • If you’re asking for the real or complex numerical value, one may ask Why? and then refer you to various numerical methods like Newton-Raphson, as @marshalcraft recommends. If you don’t care about numericals, it may be best to work purely symbolically, as in Galois Theory. Do you feel that the specific form of the solution in radicals of cubic and quartic equations has helped your understanding? Didn’t help mine. – Lubin May 26 '16 at 04:09
  • No, I'm specifically not interested in numerical values. I'm interested in symbolic solutions.

    I find the relationship between polynomials and group theory deeply interesting. But it seems arbitrary that the operations we 'allow ourselves' to use consist of +, x, /, -, and radicals. I want to know if there are other operations we can 'include', and especially if they 'mean something' on the group theory side of things.

     – Alex Kritchevsky May 27 '16 at 01:55
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    Apart from the Q&A suggested by @DietrichBurde, this one is relevant for your question: a general formula for the solution of $n$-th order polynomial equations (for every $n\in\Bbb N$) is known from the works of Belardinelli and Mellin. – Daniele Tampieri Sep 19 '20 at 08:07

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