In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be "natural" generalizations of the elliptic functions.
As a warm-up, how could one get Mathematica to return an explicit list of the symbolic solutions to the arbitrary 5th order polynomial equation
x^5 + b x^4 + c x^3 + d x^2 + e x + f == 0
and. given such a list of solutions, would one be able to plug the solutions back into the polynomial and confirm that they solve the equation?