Looking for solvable irreducible sextics to test my (& Julian's) solver, I found a paper by Boswell & Glasser which, for no apparent reason, states in the introduction an amazingly succinct quintic solver, (apparently lifted straight out of MathWorld). NeilB and I further distilled it to BJQuintic[a_, b_] := ϵ*e*E^(2*π*I*j*Range[4]/5). (Sign[#]*Abs[#]^(1/5) & /@ ({v1^2*v3, v3^2*v4, v2^2*v1, v4^2*v2}/d^2)) /. {v1 -> √d + √(d - ϵ*√d), v2 -> -√d - √(d + ϵ*√d), v3 -> -√d + √(d + ϵ*√d), v4 -> √d - √(d - ϵ*√d)} /. {d -> c^2 + 1, ϵ -> Sign[c]} /. Solve[{5*e^4*(3 - 4*c)/(c^2 + 1) == a, b == -4*e^5*(11 + 2*c)/(c^2 + 1)}, {e, c}, Rationals][[1]] E.g., In[134]:= BJQuintic[20,32] Out[134]= -((2^(4/5) (Sqrt[5]/2-Sqrt[5/4-Sqrt[5]/2])^(1/5) (Sqrt[5]/2+Sqrt[5/4+Sqrt[5]/2])^(2/5) E^((2 I j \[Pi])/5))/5^(2/5))-(2^(4/5) (Sqrt[5]/2-Sqrt[5/4-Sqrt[5]/2])^(2/5) (-(Sqrt[5]/2)+Sqrt[5/4+Sqrt[5]/2])^(1/5) E^((4 I j \[Pi])/5))/5^(2/5)+(2^(4/5) (Sqrt[5]/2+Sqrt[5/4-Sqrt[5]/2])^(2/5) (Sqrt[5]/2+Sqrt[5/4+Sqrt[5]/2])^(1/5) E^((6 I j \[Pi])/5))/5^(2/5)-(2^(4/5) (Sqrt[5]/2+Sqrt[5/4-Sqrt[5]/2])^(1/5) (-(Sqrt[5]/2)+Sqrt[5/4+Sqrt[5]/2])^(2/5) E^((8 I j \[Pi])/5))/5^(2/5) In[136]:= Chop[#^5+20*#&/@N[Table[%%,{j,5}],22]] Out[136]= {-32.0000000000000000000,-32.0000000000000000000,-32.0000000000000000000,-32.0000000000000000000,-32.00000000000000000000} (More convincing would be to get exact integers from RootReduce, but it is broken.) BJQuintic dodges the issue of putting the quintic into "Bring-Jerrard form", and relies on Mathematica to magically find rational solutions to the a and b equations. But the latter isn't hard. Just look for linear factors after eliminating e or c with a Resultant. The rest of the B&G paper breaks x^6+x^2+c*x+d sextics into two Galois cases, the first of which seems too badly misprinted to merit salvaging, despite Neil's clever reconstruction of the Mathematica from the PostScript. Why don't people computer- check their formulas? Why don't they include TeX sources including the input expressions? Even Macsyma's TeX command does that. --rwg