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Galois theory gives nice information about solvability by radicals.

I was wondering if there are any other functions we could introduce (in addition to taking radicals) that would allow polynomials with insolvable galois groups to be ``solved''. In particular, it would be nice to know whether the Galois groups of polynomials that are solvable by when we include this function satisfy some group theoretic property analogous to solvability.

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    [Elliptic functions](https://en.wikipedia.org/wiki/Elliptic_function) can be used to solve quintics. Here are some relevant posts: [1](http://math.stackexchange.com/questions/540964/how-to-solve-fifth-degree-equations-by-elliptic-functions), [2](http://mathoverflow.net/questions/133878/quintic-polynomial-solution-by-jacobi-theta-function). For sextics, [there are other functions](http://math.stackexchange.com/questions/1397297/how-to-solve-the-general-sextic-equation-with-kamp%C3%A9-de-f%C3%A9riet-functions). – 2017-02-20
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    Here are two posts on the general case: [1](http://math.stackexchange.com/questions/617125/solving-5th-degree-or-higher-equations), [2](http://mathoverflow.net/questions/23094/method-of-finding-roots-of-polynominal-equations-with-arithmetic-operations-and). – 2017-02-20
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    Thanks so much! I thought I may have seen something like it before – 2017-02-21

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