# Test Polynomials for Galois Groups

The following table lists, for each Galois group (that is, each transitive permutation group, up to conjugacy in the symmetric group of the same degree) up to degree 9, an irreducible polynomial having the given group as its Galois group over the rationals.

GroupOrderPolynomial
2T1 = S(2)2X^2-X+1
3T1 = A(3)3X^3-X^2-2*X+1
3T2 = S(3)6X^3-X^2+1
4T1 = C(4)4X^4-X^3+X^2-X+1
4T2 = E(4)4X^4-X^2+1
4T3 = D(4)8X^4-2*X^3+X-1
4T4 = A(4)12X^4-2*X^3+2*X^2+2
4T5 = S(4)24X^4-X^3+1
5T1 = C(5)5X^5-X^4-4*X^3+3*X^2+3*X-1
5T2 = D(5)10X^5-2*X^4+2*X^3-X^2+1
5T3 = F(5)20X^5-9*X^3-4*X^2+17*X+12
5T4 = A(5)60X^5-X^4+2*X^2-2*X+2
5T5 = S(5)120X^5-X^3-2*X^2+1
6T1 = C(6)6X^6-X^5+X^4-X^3+X^2-X+1
6T2 = D_6(6)6X^6-3*X^5-2*X^4+9*X^3-5*X+1
6T3 = S(3)[x]212X^6+X^4-2*X^3+X^2-X+1
6T4 = A_4(6)12X^6+X^4-2*X^2-1
6T5 = F_18(6)18X^6-3*X^5+4*X^4-2*X^3+X^2-X+1
6T6 = 2A_4(6)24X^6-2*X^5+2*X^3-X-1
6T7 = [2^2]S(3)24X^6+X^4-1
6T8 = 1/2[2^3]S(3)24X^6+4*X^5-9*X^4-51*X^3-46*X^2+8
6T9 = F_18(6):236X^6-3*X^5+4*X^4-X^3+X^2-2*X+7
6T10 = 1/2[S(3)^2]236X^6-X^5+X^4-X^3-4*X^2+5
6T11 = [2^3]S(3)48X^6-X^5-X^3-X+1
6T12 = PSL(2,5)60X^6-10*X^4-7*X^3+15*X^2+14*X+3
6T13 = F_36(6):272X^6-2*X^5+2*X^4-X+1
6T14 = PGL(2,5)120X^6+3*X^4-2*X^3+6*X^2+1
6T15 = A(6)360X^6-2*X^4+X^2-2*X-1
6T16 = S(6)720X^6-X^5+X^3-X^2+1
7T1 = C(7)7X^7-X^6-12*X^5+7*X^4+28*X^3-14*X^2-9*X-1
7T2 = D(7)14X^7-2*X^6+2*X^5+X^3-3*X^2+X-1
7T3 = F_21(7)21X^7-8*X^5-2*X^4+16*X^3+6*X^2-6*X-2
7T4 = F_42(7)42X^7-3*X^6+9*X^5-13*X^4+17*X^3-10*X^2+4*X+1
7T5 = L(7)168X^7-8*X^5-2*X^4+15*X^3+4*X^2-6*X-2
7T6 = A(7)2520X^7-2*X^6+4*X^4-5*X^3+2*X-1
7T7 = S(7)5040X^7-X^6-X^5+X^3+X^2-X-1
8T1 = C(8)8X^8+8*X^6+20*X^4+16*X^2+2
8T2 = 4[x]28X^8+2*X^6+4*X^4+8*X^2+16
8T3 = E(8)8X^8-X^4+1
8T4 = D_8(8)8X^8-4*X^6+37*X^4-66*X^2+64
8T5 = Q_8(8)8X^8+12*X^6+36*X^4+36*X^2+9
8T6 = D(8)16X^8-3*X^5-X^4+3*X^3+1
8T7 = 1/2[2^3]416X^8+10*X^6+25*X^4+20*X^2+5
8T8 = 2D_8(8)16X^8+24*X^6+126*X^4+216*X^2+117
8T9 = E(8):216X^8+2*X^4-3*X^2+1
8T10 = [2^2]416X^8-13*X^6+44*X^4-17*X^2+1
8T11 = 1/2[2^3]E(4)16X^8-X^5-2*X^4+4*X^2+X+1
8T12 = 2A_4(8)24X^8-2*X^7+X^6+X^5-X^4+2*X^3+4*X^2-16*X+16
8T13 = E(8):324X^8-3*X^6+3*X^4+2*X^2+1
8T14 = 1/2(S_4[x]2)24X^8-26*X^6+99*X^4-126*X^2+49
8T15 = [1/4cD(4)^2]232X^8-X^4-1
8T16 = 1/2[2^4]432X^8-2*X^7-2*X^6+X^5+5*X^4+X^3+3*X^2+3*X+1
8T17 = [4^2]232X^8+12*X^6+48*X^4+68*X^2+17
8T18 = E(8):E_432X^8+3*X^6+3*X^4+3*X^2+1
8T19 = [1/4eD(4)^2]232X^8+X^6+2*X^2+4
8T20 = [2^3]432X^8-4*X^6-6*X^4+4*X^2+1
8T21 = [1/4dD(4)^2]232X^8+2*X^4-4*X^2+2
8T22 = E(8):D_432X^8+13*X^6+47*X^4+44*X^2+4
8T23 = 2S_4(8)48X^8+X^7-3*X^6+X^5+8*X^4+X^2+7*X+1
8T24 = E(8):D_648X^8-X^7+X^6+X^2+X+1
8T25 = F_56(8)56X^8-4*X^7+8*X^6-6*X^5+2*X^4+6*X^3-3*X^2+X+3
8T26 = 1/2[2^4]eD(4)64X^8-5*X^5-3*X^4-5*X^3+1
8T27 = [2^4]464X^8-2*X^7+3*X^5-X^4-3*X^3+2*X+1
8T28 = 1/2[2^4]dD(4)64X^8-2*X^6+3*X^4+2
8T29 = E(8):D_864X^8-X^6-X^4+X^2+1
8T30 = 1/2[2^4]cD(4)64X^8-4*X^6-20*X^4+14
8T31 = [2^4]E(4)64X^8+12*X^6+48*X^4+72*X^2+31
8T32 = [2^3]A(4)96X^8-2*X^6-13*X^4-9*X^2+4
8T33 = [1/3A(4)^2]296X^8-2*X^7-4*X^5+12*X^4+2*X^3-14*X^2-5*X+11
8T34 = 1/2[E(4)^2:S_3]296X^8+6*X^6-4*X^5-31*X^4-12*X^3+96*X^2-26*X+29
8T35 = [2^4]D(4)128X^8+2*X^6-12*X^4-3*X^2+11
8T36 = E(8):F_21168X^8+3*X^7+20*X^4+18*X^3-18*X^2-8*X+14
8T37 = L(8)168X^8-4*X^7+7*X^6-7*X^5+7*X^4-7*X^3+7*X^2+5*X+1
8T38 = [2^4]A(4)192X^8-2*X^6-7*X^4+3*X^2+8
8T39 = [2^3]S(4)192X^8+X^4+X^2+1
8T40 = 1/2[2^4]S(4)192X^8-X^7+4*X^5-2*X^4+3*X^2-X+1
8T41 = [E(4)^2:S_3]2192X^8-2*X^6-4*X^5+4*X^3-10*X^2-8*X-1
8T42 = [A(4)^2]2288X^8-2*X^7+2*X^6-2*X^5+2*X^4-X+1
8T43 = L(8):2336X^8-X^6-3*X^5-X^4+4*X^3+4*X^2-2*X-1
8T44 = [2^4]S(4)384X^8-X^5-X^4-X^3+1
8T45 = [1/2S(4)^2]2576X^8-2*X^6+7*X^4-8*X^2-4*X+7
8T46 = 1/2[S(4)^2]2576X^8-X^7+X^5-4*X^4+5*X^3+6*X^2-2*X-1
8T47 = [S(4)^2]21152X^8-2*X^7+2*X^6+X^3+X+1
8T48 = AL(8)1344X^8+3*X^7-X^6-10*X^5-9*X^4-X^3+7*X^2+11*X+4
8T49 = A(8)20160X^8-2*X^7+3*X^5-5*X^4+2*X^3+2*X^2-X+1
8T50 = S(8)40320X^8-X^6-X^5-3*X^4+4*X^2-1
9T1 = C(9)9X^9-X^8-8*X^7+7*X^6+21*X^5-15*X^4-20*X^3+10*X^2+5*X-1
9T2 = 3[x]39X^9-15*X^7-4*X^6+54*X^5+12*X^4-38*X^3-9*X^2+6*X+1
9T3 = D(9)18X^9-3*X^8+4*X^7-5*X^6+6*X^5-X^4-5*X^3+4*X^2-2
9T4 = S(3)[x]318X^9-5*X^8-X^7+4*X^6+2*X^5+3*X^4-X^3-3*X^2+1
9T5 = S(3)[1/2]S(3)18X^9-3*X^6+3*X^3+1
9T6 = 1/3[3^3]327X^9-3*X^8-10*X^7+42*X^6-28*X^5-28*X^4+28*X^3+2*X^2-6*X+1
9T7 = [3^2]327X^9-3*X^8-21*X^7+78*X^5+69*X^4-21*X^3-39*X^2-12*X-1
9T8 = S(3)[x]S(3)36X^9-X^8+3*X^6+X^5+X^4+3*X^3+2*X^2+1
9T9 = E(9):436X^9-45*X^7-93*X^6+72*X^5+216*X^4+63*X^3-81*X^2-54*X-9
9T10 = [3^2]S(3)_654X^9+6*X^8+15*X^7+18*X^6+11*X^5+X^4-2*X^3+3*X^2-X+1
9T11 = 1/2[3^2:2]S(3)54X^9-3*X^6+3*X^3+8
9T12 = [3^2]S(3)54X^9-4*X^8+4*X^7+4*X^6-7*X^5-2*X^4+4*X^3+3*X^2-X-1
9T13 = [1/2S(3)^2]354X^9-X^6-2*X^3+1
9T14 = E(9):Q_872X^9-3*X^8+12*X^7-12*X^6+12*X^5-12*X^4+12*X^3-12*X^2+9*X-3
9T15 = E(9):872X^9-9*X^7-21*X^6+72*X^5+99*X^4-99*X^3-585*X^2+549*X+166
9T16 = E(9):D_872X^9-X^8-2*X^6-X^5+3*X^4+X^2+X-1
9T17 = 3 wr 381X^9-4*X^8-2*X^7+22*X^6-14*X^5-22*X^4+20*X^3+2*X^2-5*X+1
9T18 = [1/2S(3)^2]S(3)108X^9-X^3-1
9T19 = E(9):2D_8144X^9-3*X^8+6*X^7-18*X^6+12*X^5-24*X^4+24*X^3-12*X^2+6*X-2
9T20 = [3^3]S(3)162X^9-4*X^8+6*X^7-8*X^6+7*X^5-4*X^4+2*X^3+1
9T21 = 1/2[3^3:2]S(3)162X^9-4*X^8+6*X^7-9*X^6+13*X^5+3*X^4-24*X^3+15*X^2-5*X+5
9T22 = [3^3:2]3162X^9-4*X^8+8*X^7-11*X^6+9*X^5-3*X^4+5*X^3-4*X^2-X+1
9T23 = E(9):2A_4216X^9-3*X^8+X^6+15*X^5-13*X^4-3*X^3+4*X-1
9T24 = [3^3:2]S(3)324X^9+X^8+3*X^7+3*X^5-5*X^4+6*X^3+X+1
9T25 = [1/2S(3)^3]3324X^9-3*X^8+3*X^7+4*X^6-12*X^5+9*X^4+X^3-9*X^2+6*X-1
9T26 = E(9):2S_4432X^9-X^7-5*X^6+X^5+2*X^4+4*X^3-3*X^2-X+1
9T27 = PSL(2,8)504X^9+X^7-4*X^6-12*X^4-X^3-7*X^2-X-1
9T28 = [S(3)^3]3648X^9-2*X^8+3*X^7-X^6-2*X^5+5*X^4-4*X^3+2*X-1
9T29 = [1/2S(3)^3]S(3)648X^9-3*X^6-5*X^5+5*X^2-1
9T30 = 1/2[S(3)^3]S(3)648X^9-X^8+2*X^5-2*X^4+2*X^2-2*X+1
9T31 = [S(3)^3]S(3)1296X^9-5*X^7-X^6+7*X^5+4*X^4-2*X^3-5*X^2-X+1
9T32 = P|L(2,8)1512X^9-X^8-4*X^7+28*X^3+26*X^2+9*X+1
9T33 = A(9)181440X^9-X^8-X^7-2*X^5+4*X^4-5*X^2+1
9T34 = S(9)362880X^9-4*X^7+X^5-2*X^4+5*X^3+3*X^2-2*X-1
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