Yes:

Abel. I think we may be on different pages here... I know about the proof about the impossibility of an algebraic solution. I'm concerned with a

*non-algebraic* solution, and that is what is the citations are about--a

*non-algebraic* solution. The article at everything2 says: "Felix Klein developed in 1877 a particularly elegant method for solving quintics based on the symmetries of the icosahedron." It says that he found a solution to

*every* (I assume, it doesn't say so explicitly) quintic. It says in the very last paragraph that "Poincare among others" have extended Klein's approach to

*every* polynomial of arbitrary degree.

Let me give you my motivation for a solution in terms of non-algebraic functions (this wasn't the initial motivation, but it's the one that makes the most sense): By the fundamental theorem of algebra, there must be

*n* roots of an

*n*th degree polynomial, and given integer coefficients, these roots must be algebraic. The Abel-Ruffini theorem concerns the impossibility of using algebraic functions in doing so--but there

*must* be roots and these roots

*must* be algebraic. I've taken this as a tell that non-algebraic functions can be used to find the roots.

[snip]