Stillwell's paper `Eisenstein's Footnote' (link below) gives a lucid explanation of how Eisenstein (at the tender age of 14) obtained a solution y for y^5+y=x as an infinite series in x of the form y=x-x^5+(10*x^9)/2!-(14*15*x^13)/3!+.... Furthermore, the Bring-Jerrard Theorem asserts that a general quintic can be reduced to one of the above form, with x a radical expression of the coefficient of said quintic. http://web.a.ebscohost.com/ehost/pdfviewer/pdfviewer?sid=9dd1fb0a-49ef-42b5-... Nonetheless, I don't think this gives the whole picture since the range of convergence for x isn't that big. Namely, the above series is exactly x*4F3(1/5, 2/5, 3/5, 4/5; 5/4, 2/4, 3/4|5^5*x^4/4^4), which converges (absolutely) for |x| less than or equal to (0.8)*(0.2)^(1/4), which is approximately 0.53499. So, if one is interested in solving y^5+y=1, say, then the series above won't converge. Is there an analytic continuation of the 4F3 that would be defined for large values of x and which would still give a solution of that quintic, or is there another way to get around that? Cheers, Ahmad