Wow, 78 roots, two real. Here's one of those: Out[101]= 67/246 Sqrt[7] Sin[1/3 ArcCsc[2 Sqrt[7]]] + 245/738 Sin[1/3 ArcSin[71/98]] + 469/738 Sin[1/3 (4 π - ArcSin[13/14])] + 35/246 Sin[1/3 (4 π + ArcSin[13/14])] + 70/123 Sqrt[7/3] Sin[1/3 (2 π + ArcSin[(3 Sqrt[3/7])/2])] + 67/246 Sqrt[7/3] Sin[1/3 (4 π + ArcSin[(3 Sqrt[3/7])/2])] + ( 469 Sin[1/3 (4 π - ArcSin[(3 Sqrt[3])/14])])/(246 Sqrt[3]) - ( 35 Sin[1/3 ArcSin[(3 Sqrt[3])/14]])/(82 Sqrt[3]) + ( 245 Sin[1/3 (2 π - ArcSin[(39 Sqrt[3])/98])])/( 246 Sqrt[3]) - Sqrt[ 7/6 (-1 + Sqrt[13] Sin[1/3 (2 π - ArcSin[293/(182 Sqrt[13])])])] In[104]:= N[%%%^6 + 49*%%%, 49] Out[104]= 49.00000000000000000000000000000000000000000000000 (RootReduce wimped out.) --rwg On Fri, Aug 30, 2013 at 4:12 PM, Noam Elkies <elkies@math.harvard.edu>wrote:
Bill Gosper <billgosper@gmail.com> writes:
Mea gufa. [...]
He also had no news re the possibility of a solvable, irreducible septic trinomial. Come to think of it, I don't even recall seeing a solvable, irreducible x^6+ax+b. --rwg
Here's the full list up to 100 according to gp:
? for(a=1,100,for(b=-100,100,p=x^6+a*x+b;if(polisirreducible(p),if(polgalois(p)[1]%5,print(p," ",polgalois(p))))))
x^6 + 3*x + 3 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 3*x + 5 [48, -1, 1, "2S_4(6) = [2^3]S(3) = 2 wr S(3)"] x^6 + 8*x + 20 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 8*x + 89 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 10*x + 5 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 14*x + 35 [48, -1, 1, "2S_4(6) = [2^3]S(3) = 2 wr S(3)"] x^6 + 30*x + 93 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 40*x + 82 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 44*x + 55 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 45*x + 55 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 49*x - 49 [24, -1, 2, "2A_4(6) = [2^3]3 = 2 wr 3"] x^6 + 56*x + 62 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"] x^6 + 65*x + 13 [72, -1, 1, "F_36(6):2 = [S(3)^2]2 = S(3) wr 2"]
(I didn't use a<0 because a,b and -a,b are equivalent under x <--> -x.)
Malle's paper [M1] should give the general formulas.
[M1] G. Malle: Polynomials for primitive nonsolvable permutation groups of degree d<=15. J. Symbolic Computation 4 (1987) #1, 83-92.
NDE