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