(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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Following Hermite, here is a full solution: :[font = input; preserveAspect] HermiteQuinticSolve[t_^5 - t_ + rho_ == 0, t_] := Module[{k, b, q}, k = Tan[ArcSin[16/(25 5^(1/2) rho^2)]/4]; b = (k^2)^(1/8)/(2 5^(3/4) k^(1/2) (1 - k^2)^(1/2))* If[Re[rho] == 0, -Sign[Im[rho]], Sign[Re[rho]]]; q = EllipticNomeQ[k^2]; Map[{t -> #}&, { b ((-1)^(3/4)* (InverseEllipticNomeQ[q^(1/5)/E^((2*I)/5*Pi)]^(1/8) + I*InverseEllipticNomeQ[E^((2*I)/5*Pi)*q^(1/5)]^(1/8))* (InverseEllipticNomeQ[q^(1/5)/E^((4*I)/5*Pi)]^(1/8) + InverseEllipticNomeQ[E^((4*I)/5*Pi)*q^(1/5)]^(1/8))* (InverseEllipticNomeQ[q^(1/5)]^(1/8) + InverseEllipticNomeQ[q^5]^(1/8) q^(5/8)/(q^5)^(1/8))) , b (-InverseEllipticNomeQ[q^(1/5)]^(1/8) + E^((3*I)/4*Pi)* InverseEllipticNomeQ[E^((2*I)/5*Pi)*q^(1/5)]^(1/8))* (InverseEllipticNomeQ[q^(1/5)/E^((2*I)/5*Pi)]^(1/8)/ E^((3*I)/4*Pi) + I*InverseEllipticNomeQ[E^((4*I)/5*Pi)*q^(1/5)]^(1/8))* (I*InverseEllipticNomeQ[q^(1/5)/E^((4*I)/5*Pi)]^(1/8) + InverseEllipticNomeQ[q^5]^(1/8) q^(5/8)/(q^5)^(1/8)) , b (InverseEllipticNomeQ[q^(1/5)/E^((2*I)/5Pi)]^(1/8)/ E^((3I)/4Pi) - I*InverseEllipticNomeQ[q^(1/5)/E^((4*I)/5*Pi)]^(1/8))* (-InverseEllipticNomeQ[q^(1/5)]^(1/8) - I*InverseEllipticNomeQ[E^((4*I)/5*Pi)*q^(1/5)]^(1/8))* (InverseEllipticNomeQ[E^((2*I)/5*Pi)*q^(1/5)]^(1/8)* E^((3*I)/4*Pi) + InverseEllipticNomeQ[q^5]^(1/8) q^(5/8)/(q^5)^(1/8)) , b (InverseEllipticNomeQ[q^(1/5)]^(1/8) - I*InverseEllipticNomeQ[q^(1/5)/E^((4*I)/5*Pi)]^(1/8))* (-InverseEllipticNomeQ[E^((2*I)/5*Pi)*q^(1/5)]^(1/8)* E^((3*I)/4*Pi) - I*InverseEllipticNomeQ[E^((4*I)/5*Pi)*q^(1/5)]^(1/8))* (InverseEllipticNomeQ[q^(1/5)/E^((2*I)/5*Pi)]^(1/8)/ E^((3*I)/4*Pi) + InverseEllipticNomeQ[q^5]^(1/8) q^(5/8)/(q^5)^(1/8)) , b (InverseEllipticNomeQ[q^(1/5)]^(1/8) - InverseEllipticNomeQ[q^(1/5)/E^((2*I)/5*Pi)]^(1/8)/ E^((3*I)/4*Pi))* (-InverseEllipticNomeQ[E^((2*I)/5*Pi)*q^(1/5)]^(1/8) * E^((3*I)/4*Pi) + I*InverseEllipticNomeQ[q^(1/5)/E^((4*I)/5*Pi)]^(1/8))* (-I*InverseEllipticNomeQ[E^((4*I)/5*Pi)*q^(1/5)]^(1/8) + InverseEllipticNomeQ[q^5]^(1/8) q^(5/8)/(q^5)^(1/8)) } ] ] :[font = text; inactive; preserveAspect] The elliptic modular function InverseEllipticNomeQ function is a solution of the transcendental equation EllipticNomeQ[m] = q for Abs[q] < 1. In Mathematica V2.3 this function is built into the system. With previous versions it can be defined by: :[font = input; preserveAspect] InverseEllipticNomeQ[z_] := EllipticTheta[2, 0, z]^4/EllipticTheta[3, 0, z]^4 :[font = input; preserveAspect] Here is a numerical example for solving a quintic using Hermite's approach: ;[s] 1:0,0;75,-1; 1:1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] HermiteQuinticSolve[t^5 - t + I == 0, t] // N // Chop // Sort :[font = output; output; inactive; preserveAspect; endGroup] {{t -> -1.083954101317711 - 0.1812324444698753*I}, {t -> -0.352471546031726 + 0.764884433600585*I}, {t -> -1.167303978261419*I}, {t -> 0.352471546031726 + 0.7648844336005848*I}, {t -> 1.083954101317711 - 0.1812324444698752*I}} ;[o] {{t -> -1.08395 - 0.181232 I}, {t -> -0.352472 + 0.764884 I}, {t -> -1.1673 I}, {t -> 0.352472 + 0.764884 I}, {t -> 1.08395 - 0.181232 I}} :[font = input; preserveAspect; startGroup] NSolve[t^5 - t + I == 0, t] // Chop :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{t -> -1.083954101317711 - 0.1812324444698754*I}, {t -> -0.3524715460317265 + 0.7648844336005847*I}, {t -> -1.167303978261419*I}, {t -> 0.3524715460317262 + 0.764884433600585*I}, {t -> 1.083954101317711 - 0.1812324444698754*I}} ;[o] {{t -> -1.08395 - 0.181232 I}, {t -> -0.352472 + 0.764884 I}, {t -> -1.1673 I}, {t -> 0.352472 + 0.764884 I}, {t -> 1.08395 - 0.181232 I}} ^*)