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I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular functions or generalized hypergeometric functions and this can be done also for other equations of higher degree.

I intend that to find the solutions of a high degree equation numerical methods are more efficient than analytical methods, but I'm curious to know if there is some general method that, using some kind of special functions, can be applied to solve equations of any degree. Or any equation of different degree require different special functions? And, there is some method to identifiy the equations of degree greater than four that can be solved by radicals?

References to accessible works about this topic are wellcome.

Emilio Novati
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    The Abel-Ruffini theorem actually states exactly what is needed for an irreducible polynomial to have solutions that may be found through radicals: Let $f \in \Bbb Q[x]$ be your irreducible polynomial, and let $K\supseteq \Bbb Q$ be the field you get by adjoining the roots of $f$ to $\Bbb Q$. Then said roots can be expressed using radical expressions iff the Galois group $\operatorname{Gal}(K/\Bbb Q)$ is solvable (guess where that name came from). It just so happens that the first degree where unsolvable Galois groups are possible is $5$. – Arthur May 31 '16 at 22:18
  • That being said, determining the Galois group is no simple task, so I don't know whether that is actually what you're looking for. – Arthur May 31 '16 at 22:19
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    This summary of some MSE posts dealing with Bring quintics, Brioschi quintics, etc and the methods to solve them may be useful. – Tito Piezas III Aug 27 '16 at 01:29

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