In the classic textbook Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that “the algebraic solution of the general equation of degree n is impossible if n is greater than four. By this we mean that it is not possible to find the exact values of the roots of every equation of degree n (n > 4) by performing upon the coefficients a finite number of additions, subtractions, multiplications, divisions, and root extractions.”
That proposition is considered proven by the Abel-Ruffini theorem of 1824, as Conkwright surely knew. Tantalizingly, though, he goes on to say – without elaborating – that “the general equation of degree n has been solved in terms of Fuchsian functions.” And there, it seems, the trail ends. A web search of about an hour has yielded nothing more than various restatements of the problem.
(1) Can anyone state, in a form suitable for reduction to a computer algorithm, a solution or family of solutions of the general equation of degree n (whether based on Fuchsian functions or not)?
(2) Would your solution(s) yield theoretically exact values, or only converging approximations?
This is my first post to your site, and I apologize in advance if I've overlooked answers right under my nose. These questions have, however, stumped my math department chair. (I’m on loan to him – I normally teach humanities, but I have an engineering background and I was asked to fill in for some math courses.)