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Appendix: Example Polynomials

In this appendix we list example polynomials whose roots generate regular extension fields of Q(t), respectively number fields over Q with given Galois group of small permutation degree. The first set of examples realizes most of the equivalence types of transitive permutation groups of degree less than 12 as regular Galois groups over Q(t). (There are 301 inequivalent transitive permutation groups of degree 12.) Most of these results are new. In the second table, we collect the known explicit regular Galois realizations of primitive non-solvable permutation groups of degree at most 30 over Q(t) from the literature. For both sets of tables the results were mainly obtained by the rigidity method described in Chapter I and descent arguments.

Finally, we give example polynomials generating number fields over Q with given Galois group of permutation degree at most 12. For degree less than 12, these were either found by a random search, and then the Galois group was verified by the Galois group recognition programs in several com-puter algebra systems, or they were obtained by specializing the parametric realizations from the first set of tables. (Such specializations tend to have larger field discriminant.) The polynomials listed in this table were chosen to have minimal field discriminant known to us for the given group. The polynomials of degree 12 are taken from Kliiners and Malle (1998).

1 Regular Realizations for 'Iransitive Groups of Degree Less than 12

Here we give polynomials generating regular field extensions of Q(t) with Ga-lois groups most of the transitive permutation groups of degree less than 12. The generic formulas for polynomials with symmetric or alternating group of arbitrary degree are given separately. In all other cases the groups are num-bered according to the list in Butler and McKay (1983), so that a polynomial fn,i has Galois group the transitive permutation group of degree n denoted by Ti in loco cit.

404 Appendix: Example Polynomials

Table 1 Symmetric and alternating groups

Sn xn - t(nx - n + 1)

An { I S n (x, 1 - (_1)n(n-1)l2 n t 2) for n == 1 (mod 2)

Isn(x, 1/(1 + (_It(n-1)I2(n -1)t2» for n == 0 (mod 2)

Table 2 Degree 4

143 x4 - 2x2 + t 142 x4 + tx2 + 1 141 x4 + tx3 - 6x2 - tx + 1

Table 3 Degree 5

Is,3 Xs + 10x3 + 5tx2 - 15x + t 2 - t + 16 Is,2 x 2 (x + 1)2(x + 2) - (x - 2)2(X - l)t is,l x(x2 - 25)2 + (x4 - 20x3 - 10x2 + 300x - 95)t2 - 4(x - 3)2t4

Table 4 Degree 6

16,14 x6 - 2xs + (5x2 - 6x + 2) t 16,13 x6 - (3x - 2)2 t

16,12 16,14(x,1 - 5t2 )

16,11 x6 - (3x 2 + 1)t/4 16,10 16,13{x, 1/{t2 + 1» 16,9 16,13(x,1 - t 2 )

16,8 16,11 (x, 1/(1 - 3t2» 16,1 16,11 (x, t 2 )

16,6 16,l1(x,3t2 + 1) 16,s 16,13(x, -12t2(3t2 + 1» 16,4 16,11 (x, (t 2 + 3? /(t2 - 3)2) 16,3 X 2(X2 + 3)2 + 4t 16,2 16,3(x,3t2 + 1) 16,1 x 6 + (3x 2 + 4?(3t2 + 1)

Regular Realizations for Degree Less than 12 405

Table 5 Degree 7

17.5 (X4 - 3x3 - X + 4)(x3 - X + 1) - X 2(X - 1) t

17,4 X7 + 28x6 + 63x5 + 1890x3 + 3402x2 - 5103x + 33534 +x(x6 - 63x4 - 3402x - 5103) t + 13122 t2

17.3 see Smith (1993)

17.2 17.4 (X, (t3 - 27t2 - 9t + 27)/(3(t3 + t2 - 9t - 1))) 17.1 X7 - 21¢>7(t)x5 - 7¢>7(t)(10t3 + 5t2 - 5t - 3)x4 - 7(15t6 + 15t5

-20t4 - 27t3 - 13t2 - 6t - 13)¢>7(t)X3 - 7(12t9 + 18t8 - 30t7 - 63t6

-35t5 - 14t4 - 35t3 + 2t2 + 3lt + 16)¢>7(t)X2 - 7(t - 1)¢>7(t)(5tll

+15tlO - 5t9 - 62t8 - 93t7 - 91t6 - 126t5 - 166t4 - 113t3 - 30t2

-8t - 12)x - (6t 15 + 15t14 - 35t13 - 126t12 - 63tll + 70tlO

-91t9 - 271t8 + 131t7 + 427t6 + 126t5 - 84t4 + 175t3 + 189t2 - 29t -97)¢>7(t)

Table 6 Degree 8

/8,48 X4(X - 2)2(X2 + X + 2) - (x _1)2(x2 + X + l)t

/s,47 X8 - (4x - 3)2t

/8,46 /8,47(X, t2 + 1)

/8,45 /s,47(X, 1/(1 - e)) /S,44 XS + (4x 2 + 3) t

/S,43 X6(X2 - X + 7) -108(x -1)t

/S,42 /s,47(X, -1/(12t2(3t2 - 1)))

/S,41 (x2 _ 2)4 _ 26(2x _ 3)2t/33

/S,40 X4(X4 - 8x2 + 18) - 27t

/S,39 /8,44(X,3e) /8,38 /S,44(X, 1/(3t2 + 1)) /8,37 x8 + 6x7 + 3(7x2 + 6x + 36)(7t2 + 144)

/S,36 (fg,32(X,O) - fg,32(t, O))/(x - t)

/8,35 /8,44(X, -27t2(t - 1)/4)

/8,34 /8,41 (x, 1- t 2) /8,33 /8,41 (x, 1/(3t2 + 1)) /8,32 /8,40(X, -3t2) /8,31 /S,44 (x, 27(t2 - I? /(t2 + 3?)

/S,30 X4(X4 + 4x2 + 6) - (4x2 + 1)(3t2 + 2)2(3t2 - 1)/4 /S,29 /8,44(X, 27t2(e - 1)2/4)

/8,28 /S,44(X,27t4(t2 + 1)/4)

/S,27 /S,44(X, 27(t2 + 4)/(4(t2 + 3)3))


!S,26 !S,40(X, 27t2(t - 1)/«3t + 1)(3t - 2)2)) !S,25 see Smith (1993)

!s,24 (x2 + X + 1)4 - (2x + 1)2 t !S,23 X2(X2 + 396)2(X2 + 11) - (x2 + 4)2(X2 + 256) t !S,22 !S,41(X, 27t2/(4(t2 - 1)3))

!S,21 x 8 + 2(t2 _ l)x6 + (3t4 _ t 2)X4 + 2(t6 + t4)X2 + tS + t6 !S,20 !S,41(X, 27t4(t2 + 1)/4) !S,19 !s,41(X, 27(e + 4)/(4(t2 + 3)3)) !S,lS !S,41 (x, 27(e - 1)2/(t2 + 3/) !s,17 (x4 + 4x3 - 6x2 - 4x + 1)2 - 16x2(x2 - 1)2 t !S,16 !S,40(X, -27(t2 + 2)2/(t4(4t2 + 9))) !S,15 xS + 8x6 + 4(4t - 11)x4 + 8(t - 3)(t - 2)x2 + t(t - 3)2 !S,14 !S,24(X, 1- 3e) !S,13 !s,24(X, 1/(3e + I)) !S,12 !s,23(X, _t2)

!S,l1 !S,15(X, t2) !S,lO !S,41 (x, 2233t4(t2 + 9)(t2 + 1)/(t2 + 3)6) !S,9 !S,24(X, -27e(t - 1)/4) !s,s !S,15(X, (8t2 + 3)/(2t2 + I)) !S,7 !S,lS(X, 4/(e + I)) !S,6 !s,ls(x,2t2 + 3) !S,5 xS _ 4(t2 + 2)(t2 + l)x6 + 2(3t2 + I)(e + l)(t2 + 2?X4

-4(e + 2)2(t2 + 1)3x2e + (t2 + 2)2(t2 + 1)4t4 !S,4 (x4 _ 6x2 + 1)2 + 16x2(x2 - 1)2 t !S,3 !S,24(X,27(t2 - 1)2/(t2 + 3)3) !S,2 !S,24(X,27(t2 + 4)/(4(t2 + 3/)) !S,l XS - 4(t4 + l)x6 + 2(4t2 + l)(t4 + l)x4 - 4(t2 + l)(t4 + l)ex2

+(t4 + l)t4

Table 7 Degree 9

/9,32 x 9 + 108x7 + 216x6 + 4374xS + 13608x4 + 99468x3 + 215784x2 +998001x + 810648 + 663552 t

!9,31 x4(x + 1)3(x + 3)2 - 4/27(3x + 1)3 t /9,30 X4(X - 3)2(X3 - 3x2 - 12) + 2S t /9,29 !9,31 (x, t2) /9,2S !9,31 (x, 1/(3e + 1)) /9,27 h,32(X, (t3 - 6t2 + 3t + 1)/(t3 - 3t + 1)) /9,26 (x3 - 19x2 + 97x - 27)(x2 - 4x - 7? + 16/27x2(x - 7)t


fg,2S /9.30(X, 1/(3t2 + 1» f9,24 X6(X3 + 9x + 6) - 4(3x + 2)3 t f9,23 fg,26(X, -(43923t2 + 18225)/(3t2 + 1» fg,22 fg,24(X, 1/(3t2 + 1» fg,21 x4(x + 1)2(x + 2)2(x + 3) - 1/3sx 2(9x2 + 20x + 12) t + 1/39 t2 fg,20 fg,24(x,3t2 + 1) fg,19 (j1O,3S(X, O)(t - 1)2 - flO,3S(t, O)(x - 1)2)/(X - t)

f9,18 x6 (x + 1)2(X - 2) + 4/27(3x + 2)3 t /9.17 fg,21 (x, 2S /(3t2 + 1» f9,16 (x2 + X - 2)4(x - 4) + 2433x3 t fg,1S

fg,14

fg,13 fg,18(X, 1/(3t2 + 1» f9,12 fg,18(X, _t2)

f9,11 f9,18(X,3t2 + 1) fg,10 (x6 + 3x4 + 10x3 + 6x + 25)(x3 + 3x2 + 2) + (x - 1)(x + 2)

.(x3 - 3x2 - 6x - 1)(x4 - 7x3 + 6x2 - 13x - 14)t/9 + 18t2 /9.9 fg,16(X, t 2 + 1) f9,8 fg,16(X, t2) fg,7 (x3 + 27x2 - 9x - 27)(x2 + 3)3 - 27/4(x2 - l?x(x2 - 9)(3t2 + 49) fg,6 fg,21(X, 96(t2 - 9)2t2/«t4 - 2t2 + 49)(3t2 + 1))) fg,s fg,16(X, (t2 + 1)2/(t2 _ 1)2) fg,4 fg,16(X, 1/(3t2 + 1)2) /9.3 f9,1O(X, _(t3 + 6t2 + 3t - 1)/(t3 - 3t - 1)) fg,2 x g - 6(t2 + 3)x7 - 6x6t + 9(t4 + 9t2 + 9)xS + 24t(t2 + 3)X4 - (4t6

+69t4 + 213t2 + 81)x3 - 216t3x 2 + 12t2(3t4 - llt2 + 21)x - 8t3 fg,1 x9 _ 27¢>9(t)X7 - 54t(t2 - 1)¢>g(t)x6 + 243¢>9(t)(2t4 + t3 - e + 1)xs

+243t(t2 - 1)¢>9(t)(4t4 + 2t3 - t2 + t + 3)X4 - 81(33tB + 33e -26t6 - 6tS + 69t4 + 16t3 - 36e - 3t + 1O)¢>g(t)x3 - 2187t(t2 - 1) .(2t8 + 2t7 _ t6 _ tS + 4t4 + 3t3 _ t2 + 1)¢>9(t)X2 +729(2t3 + 1)(3t9 + 9tB + 2t1 - 14t6 + 17t4 + t3 - 9t2 - t + 1) '¢>9(t)X + 243¢>9(t)(36tI3 + 18tI2 - 60t11 + 30t1O +64t9 - 8It8 - 9t1 + 87t6 - 36tS - 54t4 + 2It3 + 15t2 - 3t - 1)


Table 8 Degree 10

flO ,43 XlO - (5x - 4)2 t

flO,42 /10,43 (X, I/(e + 1» flO,41 flO,43(X,I - t 2)

flO,40 flO,43(X, -2Ot2(5t2 - 1»

flO,39 x lO _ 55(X2 + 4) t flO,38 flO,39(X, I/(t2 + 1» flO,37 (x2 _ 4)5 - 55x 2 t

flO,36 flO,39(X,I - 5t2)

/10,35 X10 _ 2x9 + 9x8 - 729(x - 1)2 t flO,34 flO,39(X,4(e + t - I)2/(5(t2 + 1)2» flO,33 (x - 2)2(X2 + X - 1)4 - (380X6 - 784x5 + 300x4 + 360x3

-315x2 + 60x + 4) t + 4(5x - 4)2 t2 flO,32 /10,35(X, t 2)

flO,31 flO,35 (X, 1/(2t2 + 1» flO,30 flO,35(X,1 - 2t2)

/10,29 XlO + lOx6 - 5tx4 - 15x2 - t2 + t - 16 flO ,28 (x - 2)(x - I)(x4 + X3 + 6x2 - 4x + I)(x2 + X - I?

+(lOx3 - lOx2 + I)(4x5 - 20x2 + I5x - 2) t +(5x - 4)(8x5 - 40x2 + 35x - 8) t2

flO,27 flO,33(X, -t/(4(t2 - 1)))

flO,26 flO,35(X, (t2 - 2? /(t2 + 2)2) flO,25 /10,39(X, 28e /«t4 + 6t2 + 25)(e + 1)4» flO,24 flO,29(X, t(t - 8)/(e - 1»

flO ,23 X2(X4 _ 25)2 + (X8 - 20x6 - lOx4 + 300x2 - 95)t - 4(x2 - 3?t2 flO,22 XlO + 55(X2 + 256)4t flO,21 (X 2 + 9x + 24)2(X2 + 4x + 64?(x2 - 6x + 144) - 55x4(X + 8? t/4 flO,20

/10,19 (X2 + I)4(x2 + 16) - 5(x7 + llx5 - I5x4 - 5x3 + 38x2 - I5x - 7) t +(x5 + lOx3 - I5x - I5x2 + 28) t2

flO,18 /10,28(X, t 2) flO,17 flO,16 flO,23(X, t 2 - 95/36) flO,15 /10,23 (X, 95/(t2 - 36» flO,14 /10,23 (X, t 2) = /5,1 (X2, t) flO,13 (X2 - 5)5 _ 55(X2 + 5x + 6)4 t/4 flO,12 /10,22(X, 1/(1 - t2» /10,11 /Io,22(x, 1 - 5t2)

/10,10 /IO,21(X,4(3t2 + 32)/(4t2 + 1» /10,9 /10,21 (x, -4(5t2 - 32»


flO,8 flO,23(X, (7t2 - 24t + 7)2/(36(t2 - I?»

flO,7 iIO,13(X,1 - 5t2)

flO,6 iIO,19(X, -4/(5t4 + 5t2 + 1))

flO,5 iIO,22(X, -4t5(t - 10)/(55 (t2 + 2t + 5)))

flO,4 iIO,13(X, -4t5(t - 1O)/(55(tz + 2t + 5»)

!to,3 flO,22 (x, -(l1t2 + 4t - l1)(tz + 4t - 1)5/(55(t2 + 1)2(t2 - 1)4»

flO,2 X10 _ 2(tz - 125)x8 + (t2 - 125)(t2 - 4t - 65)x6 - 4(t2 - 125)2 ·(t - 1O)x4 + 4(t2 - 14t + 25)(t2 - 125)2x2 - 64(2t - 25)(t2 - 125)2

!tO,l x 10 _ 20¢1O(t)X8 + 1O(7t4 - 7t3 + 17t2 - 17t + 12)¢1O(t)X6

-25(4t8 - 8e + 12t6 - 16t5 + 25t4 - 46t3 + 67t2 - 38t + 9)¢1O(t)X4

+5¢1O(t)(13t12 - 39t11 + 18t10 + 50t9 - 125t8 + 376t1 - 453t6

-214t5 + 1050t4 - 1125t3 + 613t2 - 164t + 18)x2 - ¢lO(t) (-1 - 3t + 32t2 - 36t3 - 10t4 + 34t5 - 13t6 - 8t1 + 4t8)2

Table 9 Degree 11

f11,6 (fM12(X,0)(2t - 1)2 - fM12(t,0)(2x -In/(x - t) f11,5 Xll _ 3x10 + 7x9 - 25x8 + 46x7 - 36x6 + 60x4 - 121x3 + 140x2

-95x + 27 + x 2(x - I? t

f11,4

f11,3

f11,2

fll,1 xll - 55¢11(t)x9 - 11(30t5 + 15t4 - 30t3 - 25t2 - 4t + 3)¢11(t)x8

-11(90t lO + 90t9 - 240t8 - 350t1 - 229t6 - 97t5 + 35t4 + 13t3

-42t2 - 42t - 75)¢11(t)X7 - 11(168t15 + 252t14 - 840t13 - 1750t12

-1218t11 - 242t10 + 880t9 + 1265t8 + 880e + 836t6 + 572t5 + 437t4

+430t3 + 224t2 + 78t - 36)¢11 (t)x6 - 11(21Ot20 + 420t19 - 1680t18

-4550t17 - 2723t16 + 2118t15 + 7971t14 + 11976t13 + 9282t12

+6555t11 + 6523tI° + 5466t9 + 6103t8 + 4089t1 - 422t6 - 2128t5

-1887t4 - 722e + 355t2 + 508t + 452)¢11(t)x5 - 11(180t25 + 450t24

-2100t23 - 7000t22 - 3080t21 + 10615t20 + 27060t19 + 40865tl8

+32857t17 + 10109t16 - 2398t15 - 10128t14 - 6994t 13 _ 882t 12

-14413t11 - 33099t lO - 42438e - 36861t8 - 18117e - 550t6

+6589t5 + 2640t4 - 1063t3 - 958t2 - 648t + 117)¢1l(t)X4

-11(105t30 + 315t29 - 1680t28 - 6650t27 - 1659t26 + 20003t25

+44905t24 + 64445t23 + 44116t22 - 34353t21 - 102124t20 - 135499t19

-138713tl8 - 92626t17 - 79067t16 - 119189tl5 - 147399t14

-166843tl3 - 136359t12 - 38237tll + 44396t10 + 74899t9 + 52267t8

+203lt7 - 22096t6 - 1205lt5 + 3115t4 + 7001e + 1543t2 - 1896t


-1160)4>u(t)x3 - 11(40t35 + 140t34 - 840t33 - 3850t32 - 154t31 +19008t30 + 3960Ot29 + 49203t28 + 1452Ot27 - 120945t26 - 280357t25

-348952t24 - 314514t23 - 145540t22 + 29359t21 + 33825t20

-27126t19 - 75933t18 - 85096t17 + 57717t16 + 276738t15+ 420112t14 +438965t13 + 296100t12 + 50632tU - 97383t10 - 69608t9 + 16104t8

+68277t7 + 54527t6 - 3025t5 - 25355t4 - 7986t3 + 3117t2 + 2302t -84)4>u (t)X 2 - 11(9t40 + 36t39 - 240t38 - 1250t37 + 227t36

+9128t35 + 17905t34 + 16150t33 - 12716t32 - 122980t31 - 290048t30

-377822t29 - 31155lt28 + 1083t27 + 489620t26 + 744371t25

+66292lt24 + 433805t23 + 172463t22 + 209836t21 + 561407t20

+810964t19 + 909892t18 + 777874t17 + 28980lt16 - 197823t15

-430310t14 - 356065t13 - 11405t12 + 252280tU + 23013lt lO

+47388t9 - 93665t8 - 90187t7 - 24467t6 + 19479t5 + 19576t4

-4165t3 - 586lt2 + 1587t + 999)4>u(t)x - (IOt45 + 45t44 - 330t43

-1925t42 + 792t41 + 19448t40 + 36036t39 + 1376lt38 - 83787t37

-449020t36 - 113895lt35 - 1569333t34 - 1270152t33 + 131912t32

+3449677t31 + 7101292t30 + 8022157t29 + 6359584t28 + 2562879t27

-1238875t26 + 266530t25 + 479238lt24 + 7758954t23 + 9292575t22

+6341588t21 + 10748lt2o - 3610200t19 - 4999456t18 - 3552868t17

+1494614t16 + 4899972t15 + 3760834t14 + 62019lt13 - 2831935t12

-4464948tll - 2695792t10 - 670956t9 - 36608t8 + 32528lt7

+187935t6 + 13585t5 + 170786t4 + 81906t3 - 42372t2 - 19548t +243)4>11 (t)

2 Regular Realizations for Nonsolvable Primitive Groups of Degree up to 30

Here we collect the known regular realizations for primitive non-solvable permutation groups of degree d with 12 ~ d ~ 30. Simple groups in this range for which no polynomial over Q(t) is known at present are the five groups L2(16), L3( 4), M23, L2(25) and L2(27). The polynomials were taken from Hafner (1992), Malle (1987, 1988a, 1993a), Malle and Matzat (1985), Matzat (1987), Matzat and Zeh-Marschke (1986). In addi-tion we present the polynomial with Galois group Z16 from Dentzer (1995a).

Regular Realizations for Nonsolvable Primitive Groups 411

Table 10 Primitive groups

M12 X12 + 44xll + 754x lO + 6060x9 + 18870x8 - 28356x7 -272184x6 - 57864x5 + 1574445x4 - 92960x3 - 1214416x2 + 1216456x - 304119 - 492075(2x - 1)2 t

PGL2(1l) (x3 - 66x - 308)4 - 9t(llx5 - 44x4 - 1573x3 + 1892x2

+57358x + 103763) - 3t2(x - 11) L2(1l) /PGL2(11) (x, 2835/(llt2 + 1» L3(3) (x3 - 18x2 + 54x - 108)(x4 + 8x3 - 108x2 + 432x - 540)

·(x6 - 6x4 + 64x3 - 36x2 + 216) _(3X4 - 28x3 + 108x2 - 216x + 108?(x4 + 8x3 + 108) t

PGL2(13) (x3 - x 2 + 35x - 27)4(X2 + 36) - 4(x2 + 39)6(7x2 - 2x + 247) t/27 L2(13) / PGL2(13)(X, 1/(39t2 + 1» PGL2(17) (x3 _ 7x2 + 5x - 2)6 _ (X 17 _ 17x15 + 34x14 + 85x13 - 408x12

+289x11 + 1190xlO - 2907x9 + 1462x8 + 3281x7 - 5780x6 +3196x5 + 238x4 - 646x 3 - 68x2 + 120x - 16)t + t2

L2(17) /PGL2(l7) (x, 28 17/(t2 -17» PGL2(19) (x 5 + 26x4 + 69x 3 + 108x2 + 68x + 16)4 _ (x 19 - 38x17

-38x16 + 513x15 + 1064x14 - 2299x13 - 9538x12 - 5358xll +24358x lO + 55081x9 + 35416x8 - 40204x7 - 105374x6 -98496x5 - 41040x4 + 3648x3 + 11552x2 + 4352x + 512)t + t2

L2(19) fpGL2(19) (x, 2819/(t2 + 19» L3(4).22 (fAut(M22)(X,0)(t2 - t + 3)11 - (fAut(M22)(t, 0)(x2 - x + 3)1l)/(t - x) Aut(M22 ) (5x4 + 34x3 - 119x2 + 212x - 164)4(19x3 - 12x2 + 28x + 32?

_222(X2 _ x + 3)11 t

M22 fAut(M22)(X, 1/(llt2 + 1» M24 see Granboulan (1996)

PGL2(23) (x8 + 3x7 + 37x6 - 24x5 + 121x4 + 333x3 + 429x2 + 216x +36)3 _ (2X24 + X23 _ 322x22 + 1219x21 + 1863x2o + 4094x 19 +99084x18 + 197501x17 + 877910X16 + 1337726x15 +3132117x14 + 8697795x13 + 15394935x12 + 16590866xll +4182642x lO + 6982731x9 + 36934642x8 + 4308560lx7

+13510591x6 - 9423054x5 - 10152936x4 - 4024080x3

-824688x2 - 85536x - 3456)t + (X24 - 7X23 + 69x22 - 460X21

-1564x2o - 3289x19 + 11017x18 + 19159x17 - 20792x16 -269307x15 - 650440x 14 - 547124x 13 + 609937x 12 +2106294xll + 2682306x10 + 1410682x9 - 856612x8 -1557215x7 - 609132x6 + 135079x5 + 225814x4 + 113436x3 +33764x2 + 5904x + 496)t2 - (X 23 + 23x20 + 23x19 + 23x18 +161x17 + 368x16 + 529x15 + 575x14 + 161Ox13 + 3036x12 +2668x11 + 2300x10 + 3542x9 + 5428x8 + 2599x7 - 1748x6 -1265x5 + 345x4 - 598x2 - 252x - 16)t3 + t4


L2(23) jPGL2(23)(X, (23 - 33 t 2)/(t2 + 23)) U4(2).2 (X3 + 6X2 - 8? - 24312x6(X2 + 5x + 4)4(X - 2) t

U4(2) (x3 + 6x2 - 8)9 - 24312x6(x2 + 5x + 4)4(X - 2)(3t2 + 1)

86(2) (x4 - 10x2 - 8x + 1)7 - X3(X2 + 3x + 1)5 t

U3(3).2 (X6 - 6x5 - 435x4 - 308x3 + 15x2 + 66x + 19)4 '(X4 + 20x3 + 114x2 + 68x + 13) - 2239(X2 + 4x + 1)12(2x + l)t

U3(3) j U3(3).2(X, 1/(t2 + 1)) PGL2(29) (x5 _ 7x4 + 8x3 - 17x2 + 9x _ 6)6 - t(x29 + 29x26 - 29x25

+29x24 + 290X23 - 638x22 + 899x21 + 464x20 - 4118x19

+8323x18 - 9686x 17 - 899x16 + 20532x 15 - 46197x14

+55477xI3 - 36801x12 - 8584xll + 66874x lO - lOO601x9

+105560x8 - 73602x7 + 34017x6 - 2349x5 -11745x4

+ 10962x3 - 6264x2 + 1944x - 432) + t2

L2(29) jPGL2(29)(x,223329/(tl - 29))

Table 11 The cyclic group Z16

Z16 X16 - 24¢>16(t)xI4 + 24(16t6 - 14t4 + 6t2 + 5)¢>16(t)x I2 - 26(24t 12

_28t10 + 6t8 + 36t6 - 3lt4 + 13t2 + 2)¢>16(t)X10 + 25(128t 18 - 120t16

-144t14 + 560t 12 - 488t lO + 144t8 + 164t6 - 136t4 + 56t2 + 1)¢>16(t)X8

_28(16t22 + 16t20 - 120t18 + 208t16 _ 108t14 - 64t 12 + 164t lO - 128t8

+73t6 - 20t4 + 3t2 + 2)tl¢>16(t)X6 + 28(64tl4 - 192t22 + 208t2° +80t18 _ 432t 16 + 520t14 - 316t12 + 112t lO + 18t8 - 66t6 + 67t4

-26t2 + 5)t4¢>16(t)X4 - 210(32t22 - 112t20 + 160t18 _ 72t16 - 84t 14 +144t12 - 86t lO + 28t8 - 17t6 + 17t4 - 7t2 + l)t6¢>16(t)X2

+28 (8t lO - 16t8 + 12t6 - 4t2 + 1)2t8¢>16(t)

3 Realizations over Q for Transitive Groups of Degree up to 12

This last set of tables contains polynomials generating field extensions of Q with transitive Galois group of degree less than thirteen.

Degree 2

Degree 3

Realizations over Q for Transitive Groups 413

Degree 4 T~ Sd X4 - X + 1 T AA i4 - 2x3 + 2xz- + 2 T Dd X4 - x 3 - x-Z-+ x + 1 T V.1 X4 - x-Z-+ 1 T, 4 X4 + x:3 + x 2 + X + 1

Degree 5 T. S. x 5 - x 3 - x 2 + X + 1 T A.~ i 5 + X4 - 2X2 - 2x - 2 T FJfl is + x4 + 2i:f + 4x-z-+ x + 1 T D. x 5 - x:3 - 2X2 - 2x - 1 T, 5 x 5 + X4 - 4x3 - 3x2 + 3x + 1

Degree 6 Tlfi Sr. Xo - X4 - x--:f + X + 1 Tg Ar. Xo - 2X4 + XI - 2x - 1 T14 PGL2(5) x 6 + 3x4 - 2x3 + 6x2 + 1 Tp F.D.1 xo+xo-x"'I-x+l T12 L2(5) Xo - 2xo + i4 + 2x--:f + 2x:.! + 4x + 1 T11 2 X Sd x 6 + x 5 + 2x4 + 2x--:f + 2x"'I + 2x + 1 T'f F.4 x6 + i 5+ i4+ x:f - 4x"'I + 5 To F.i2" XO - 2X4 - 4x:3 + 6x2 + 4x - 1 Ts S4/4 x 6 + 2x5 - X4 - 4x3 + 7x2 - 4x + 1 T7 s47v4 XO - x:.!-l T" 2 x Ad XO + Xo - 2i~ + 2x - 1 T~ 3 X S? x 6 + x4 - x 3 - 2x2 + X + 1 Td Ad x 6 + x4 - 2X2 - 1 T'>. Dr. xO +0-2x--:f+ x"'I- x +l T? S ... :TO + XO + 40 + x:3 + 2x:.! - 2x + 1 T, 6 :TO+XO+0+x--:r+ x"'I+ x +l

Degree 7 T7 S7 X 7 + x6 - x5 + x 3 - x 2 - X + 1 Tr. A7 x7 + 2x6 - 4i4 - 5x3 + 2x + 1 T5 L 3(2) x'" - XO - 3x5 + X4 + 4x :.J - x:.! - X + 1 Td F.1? x'" + 2xo - 2xo - x4 + 6i:f - x + 4 T1 F?l x'" - 8xo - 2X4 + 16x:f + 6x"'I - 6x - 2 T D7 X 7 + Xo + 2xT + 4x-:3 + 2x + 1 T, 7 x-" - XO - 12i5+ 7x4+ 28x-:3 - 14x2 - 9x - 1

Degree 8 T"n S" xl! - 2x5 - X4 + 3x:3 - X + 1 T.1o A" xl! - X 7 + 2Xfi - 3x5 + 2X4 - x:.J + 3x:.! - 3x + 1 T4S 23 .L3(2) xII + 2x o - 5xo + 6x:3 - 5x + 2 Td7 8.1/2 XS - 2x5 + i4 + 3x3 - 2i2 - X + 1


TA

Realizations over IQ for Transitive Groups 415

Degree 9 T34 89 x!i - Xli + 2x5 + X4 - 3xd + x:.l + x-I T33 Ag x!i + 2x' - X4 - 4xd + 2x :.l + 2x - 1 T32 rL2(8) Xli + x·1 + 2x tJ + 4xJ - x:.l + X + 1 T3I 83 183 Xli - 2XIS - Xl + 2xb - XD - X4 + x d + x:.I - 1 T30 Xli + 2x tJ - 4X4 + 4xJ - 4x :.l + x-I T2'J x!i - 2XlS + 2x7 - 4Xli - X4 + 5x3 - 3x + 1 T21'1 8:d3 Xli - 2XIS + 3x 7 - x b - 2x5 + 5X4 - 4xJ + 2x - 1 T27 L 2(8) x!i - 12xo - 18x" + 36x2 - 27x - 128 T26 3:.1, GL2 (3) x!i - x 7 + 5xli + x 5 - 2X4 + 4xJ + 3x :.l - x-I T')r; x!i - 3xb + 9x" - 9x4 - 27 x J + 9x + 1 T24 x!i - xl! + 3x 7 + 3x5 + 5x4 + 6x3 + x-I T23 3:.1, SL2 (3} x9 - 3x l! + Xo + 15x t> - 13x4 - 3x" + 4x - 1 T22 x!i - xl! + x7 - 2x5 + 5x4 -llxJ + llx:.l - 2x - 1 T2I x!i + 2x l! + 2Xli + 2x5 + 6x2 - 4x + 2 T20 3/83 x9 + 2Xli - 4x5 + 7x4 - 8x3 + 6x2 - 4x + 1 T}g Xli - XIS + 4x' + 12xo + 8xtJ - 24x d - 24x2 - 12x - 4 TIR Xli - x d -1 TI7 313 Xli - 4x lS - 2Xl + 22xo - 14x5 - 22x4 + 20xJ + 2x:.l

-5x+ 1 Tu; 32,D4 x9 - 2x 7 - 3xo + x" + X4 - 3xJ + x-I T I5 3:.1,8 Xli - 72x' + 1464x5 - 960x4 - 8928xJ + 13440x2

- 2064x - 2560 114 32,Q4 x9 - 12x5 + 132x - 128 Tl3 x!i - x b - 2Xd + 1 T12 x!i - 2XlS - 4x7 + 9x" + 17x" + I2x d - X-.: - 4x - 1 Til 3:.1,6 x!i - 3Xli + 3xJ + 8 TIn 9,6 X9 - X7 - 2Xli + 3x5 + X4 + 2Xd - x:.! + X - 3 T9 3:.1.4 Xli - 3x l! + 3x7 - 15xIi + 33xo - 3x" + 24x3 + 6x2 - 4 TB 8J Xli - XIS + 3xo + x 5 + X4 + 3xJ + 2x:'! + 1 T7 3:.1,3 Xli - 3x1l - 15x7 + 51xli + 39xo - 219x" + 8Ix" + 204x2

-132x - 8 n 9.3 x!i - 21x7 - txb + 126xtJ + 42x4 - 273xJ - 63x2

+189x + 7 Ts 32 ,2 Xli - 3xo - 3xJ - 1 T4 83 X 3 Xli - 6xo - 16xJ - 8 T1 Dg x!i + XIS + 3xb + 3xd - 3x :.l + 5x - 1 T2 32 x!i _ 15x·' + 4xo + 54x t> - 12x4 - 38xd + 9x:.! + 6x - 1 Tl 9 x!i + XIS - 8X' - 7xo + 21xt> + 15x4 - 20xd - lOx"

+5x+ 1


Degree 10 T45 S10 X lU + Xli - X~ - X7 - XO - 2X4 - x;j + 2x :t + 2x'+ 1 T44 A10 x lU - 2X'1 - x b + X4 - x;j + x:t - X + 1 T4~ Sd2 x lU - x~ - XO + 2X4 + 2x;j - 2x :l - X + 1 T42 x lU + lOxo - 8x5 - 25x :l + 40x - 16 TAl x lU + Xli - xl! - 2X7 - Xo - X5 + 3x4 + 2x;j + X:l - 3x + 1 T40 A5/ 2 x lU + Xli - xl! - x7 - 2xo + 2x;j + 3x :l + X + 1 T39 2/ S5 x lU + xl! - x 7 - x 5 - x;j + x:l + 1 T38 x lU + Xli - xl! - 2x7 - 3xo + x 5 + 3x4 - 2x;j + x:l + x-I T37 24,S5 x lU - 4xo + 3x4 + 2X2 - 1 T36 2/ A5 x lU + 3xB - 2x7 + 7x6 - 2x5 + 7x4 - 2x;j + 3x2 + 1 T35 PfL2i91 x lU - 2Xll + 9x8 + 2916x:'! - 5832x + 2916 T34 24.A~ x lU + 2xo - 4X4 - 3x :l - 4 T33 (5.4) / 2 x lU - 2Xll + 6x l! - 8x7 + 12xo + 24x4 + 68x :l - 24x + 40 T19 Sfi x lU - 2Xll + XIS - 9x:l + 2x - 1 T3l MlO x lU - 2Xll + 9x lS - 54x :l + 108x - 54 T30 PGL2(9) x lU - 2Xll + 9xl! - 7x :l + 14x - 7 T29 2/ (5.4) XW + Xli - x~ - 2x7 - Xo + 3xb + X4 - 2x;j + x:l + x-I T2R x lU - 10x l + lOxti + 36xo + 50x4 - lOx;j - 1 T27 x lU + 3xti - 2xo + x:t + 2x + 1 T26 L2(9) x lU - 4Xll + 12x7 - 12xti + 12x4 - 48x;j + 64x + 32 T2" x

lU - 2Xll - 2Xi + 4x o + 2X4 + lOx.l + 2X2 + 8x + 2 T2, 24.5.4 x lU - 2x~ - 4xti + 8x :t - 4 T23 2/ (5.2) x lU - 2x8 - Xi + 3x l:> + 2x5 - 2X4 - 2x;j + 2X2 + 3x + 1 T22 S5 X 2 x lU - Xli + 2x l! - X 7 + 2xti - 2x5 - 2x;j + x:l + 1 T2 D5/ 2 x lU + Xo - 2x5 - X4 + 3x :l - 2x + 1 T20 5:l.Q4 x lU + 20x lS + 70xo + 42x b - 425x4 + 420x;J + 275x2

-630x + 436 T19 52 .D4 x lU - 10xlS + 35x6 - 2x5 - 50x4 + 10x;j + 25x2 - lOx + 2 T18 5:l.8 x lO + 60xo - 208x5 + 850x2 - 8000x - 4672 T17 x lU + 2xo - 7 Tl~ x lU + 7x~ + 17xIJ - 31x4 - 40x:t + 127 THi 24.5.2 x lU - xl:> - 2X4 - 2x:t - 1 Tl 2/5 x lU - Xli - 9x7 + 2xti + 29xo - 27x4 - 4x;j + llx:t - 4x + 1 T13 S5/D6 x lU - Xli - x~ + 3xti - XO - 2X4 + 3x;J - x:t - X + 1 T12 S5JA4 x lO + 2x9 + 3xB - x 6 - 2x5 - X4 + 3x :l + 2x + 1 Tl A5 X 2 XW + XIS - 4x2 + 4 T 5:l.4 x lU + 36x5 - 176 T9 52 .2:l x lU - Xli - 5x lS + llxo + 4x5 - lOx4 + 25x2 + 5x - 5 Ts 24.5 x lU - 4x8 + 2xIJ + 5x4 - 2X2 - 1 T7 As x lU - xll - 4Xl - 5x l:> - 8xO - 3x4 + 4x;j + 4x2 + 4 T6 5/2 x lU - Xli + 3xi - 3xIJ + Xo + 5x4 - x.l + 2x:t + 3x + 1 Ts 2x5.4 x lU - 2x~ - xti + 5x4 - 5x :t + 3 T4 5.4 x lU + 22x o - 4 T1 DIn x lU - 2Xll + 2x~ - 2x'l + 2xti - XO + 3x4 - 4x;J + x:t + 1 T2 Dr. XW - 2x~ + txo + 41x4 + 103x :l + 47 Tl 10 XW + Xli + XIS + x7 + XO + XO + X4 + x;j + x 2 + X + 1


Degree 11 T8 8 11 XU + x lO - Xli - XIS + 2xl> - 3x4 - X;S + 2x + 1 T7 All XU - Xli + £f - xl> + 2x o + X4 - 2x;s - x-I T6 Mll XU - 4x lU - 75x7 + 540xti - 210x5 - 22500x;s + 56000x:.!

-82100x - 57600 T5 L2 (11) XU - 22xll - 88x!! + 572x7 -1408xti + 2816x5 + 1056x4

+ 1232x3 + 352x2 - 128 T F llO XU - 3 T3 F55 Xll - 33xll + 396x' - 2079xo + 4455x;s - 2673x - 243 T2 Dll XU - XlU + 5xll - 4x lS + lOx'( - 6xl> + 11x o - 7x4 + 9x;s - 4x~

+2x + It TI 11 XU + x lO - lOx9 - 9x lS + 36x' + 28x6 - 56x5 - 35x4 + 35x3

+15x2 - 6x-l

t: Taken from Bruen, Jensen, and Yui (1986).

Degree 12 8 1 ') Xl~ - X + 1 Al2 xl2 + 30x:.! + 120x + 121 T299 Xl:.! - x 9 - x7 - Xo - X4 + x,j + x~ + X + 1 T29R Xl2 - 72x~ - 120x - 50 T')Q7 Xl~ - 24xi + 144x~ + 200 T296 Xl~ - 12xll + 36x lU + 6251xti - 37506x5 + 9768751 Ml2 Xl:.! - 375x!! - 3750xti - 75000xJ + 228750x2 - 750000x + 1265625 T294 Xl:.! - Xll + x lO + Xli + X + XO + 3x4 - x,j + x2 + 1 T293 Xl:.! - x 6 - X:.! - 1 T292 Xl2 _ 3Xll + 5xll - 3x lS + 3x' + 2xl> - 6xo - 3x4 + 1 T291 xl:l - 12x9 - 9x!! - 64x3 - 144x2 - 108x - 27 T290 Xl:.! - 2XlU - 6x9 + X!! + 8x7 - lOx5 - 7X4 + 20xJ + 2x:'! - 2x - 4 T289 Xl:.! - Xll - x lO - x7 - x 5 + 3x4 - x 3 + 2x:.! - X + 1 T288 Xl:.! - 4Xll + 4x lU - 50x4 + 120xJ - 112x:'! + 48x - 8 T287 Xl:.! - 3x lU - 3x!! + 2X4 + 2x:'! + 2 T2RIl Xl:.! - xl> - 3x4 - 1 T2Rr; Xl~ - XlV - 2X4 + 1 T2R4 Xl~ - 12xll - 9x!! + 64x;s + 144x:'! + 108x + 27 T2B3 Xl:.! - 8Xll + 24xo + 144xo + 96x;s + 144x:'! + 48 T282 Xl:.! - 7x lU - 8x9 + 7x!! + 16x7 - xti - 16xo - 16x4 - 16x;s + 24x~

+40x - 8 T2Rl XU - 12x lU + 54x lS - 129x l> + 207x4 - 70x;j - 189x~ + 210x - 153 T 2RO Xl~ - 4Xll + 6x lU - 2Xll - 5x!! + 6x" - 4x o + 2X4 + 2 T27fl Xl~ - 4xu + 4x lU + 4x7 - 6xo - 4x" + 36x~ + 36x + 9 T27R Xl~ + 20x lS - 80xl> + 50x4 - 320x;s - 912x:'! + 1280x + 800 T277 Xl~ + 3xl> + 3x:.! + 4 T271l Xl~ + 192xo - 288xo + 108x 'l + 256x;s - 576x:'! + 432x - 108 T275 XU - 4xu - 6Xll + 55x!! + 18x'( + 104x l> - 24x o + 6x4 + 16x,j

+ 104x2 + 48x + 16 T274 XU + XU + XlU - Xli + xl> + 3xo + x J - X + 1 T273 Xl2 - 12xll + 9x lS + 192x;J - 432x~ + 324x - 81


Ml1 Xl:.! - xlI - 16xlU + 15xll + 145x l:! - 8Xl - 392xO + 88x5 + 415x4 -255x3 - 64x2 + 89x - 41

T271 Xl:.! - 135xll - 180x7 + 399x6 + 918x5 + 693x4 + 352x J + 216x :.l +96x + 16

Tno Xi:.l - 2x l:! - 4x:.! + 8 T 269 Xi:.l - 4Xll + 4x lU - 5xll + 40x7 - 40xti - 40x5 + IOOx4 + 40x:.!

+240x + 160 T2flR Xi:.l + 4Xll - 3x l:! - 64x.1 + 144x :.l - 108x + 27 T2fl7 Xi:.l + 12x lU - 8Xll + 54x l:! - 48x·( + 132xti - 72xtl - 33x4 - 32x.1 + 8 T2flfl Xi:.l - 8xll - 9x l:! + 24xo + 36xtl - 81x4 - 32x.1 - 36x :.l + 16 T265 xn + 36x lU - 24xll + 333x l:! - 288xl + 8lOxo - 72xtl - 486x4 + 80xJ

+162x2 - 72x + 9 T264 Xl:.! - 4Xll + 6xlU - 3x9 - 2x l:! + 3Xl - 2x5 + X4 + XJ - x'/. + 1 T263 xn - 162x4 - 432xJ - 432x2 - 192x - 32 T262 Xl:.! - 72XIl - 96xl + 184xo + 432x t> + 369x4 + 280xJ + 216x'/.

+96x + 16 T2fll xn - 4Xii + 6x lU - 4Xll + xl:! + 1 T2flO XU -3x :.l +3 T2F.9 Xi:.l - 12x lU + 54x l:! - 110xti + 93x4 - 4xJ - 18x'/. + 12x - 8 T258 Xi:.l - 2Xll + 2x J + 3 ~57 Xi:.l + 3x l:! - 2xti + 6x4 + 1 T256 Xi:.l - 3x l:! + 6x4 - 8x'/. + 2 T255 Xl:.! - 2Xll - 6xti + 9x4 - 1 T254 Xl'/. - 12xll - 2XlU + 316xll + 381x l:! - 2760xl - 11742xo - 26260x5

-42490X4 - 52184x3 - 48664x2 - 32640x - 11943 T253 Xl'/. - 4x !l - 3x l:! - 32xti - 48x5 - 18x4 + 64xJ + 144x:'! + 108x + 27 T252 xn - 12x !l + 27xll + 12x6 - 36x5 + 27x4 - 16x.1 + 36x :.l + 9 T251 xn + 48xo - 72x5 + 27x4 + 64x3 - 144x2 + lO8x - 27 T250 Xl'/. + XlU + Xll - Xti + 2 T24!1 xn - 12x lU + 54xll - 108xo + 81x4 - 8x.1 + 24x + 8 T24R Xi:.l + 324xo - 648xtl + 675x4 - 744x.1 + 648x :.l - 288x + 48 T247 Xi:.l - 8Xll + 24xo + 162x4 - 32x J + 16 T24fl XU + 81x4 - 216x.1 + 216x :.l - 96x + 16 T245 Xl2 - 12x lO - 54x l:! - 72x7 + 96xti + 9x4 + 200xJ + lO8x'/. - 4 T244 x 1 :.1 + 12x lU - 6Xll + 54x l:! - 54x·( + 124xo - 162xb + 177x4 - 178xJ

+144x2 - 48x + 16 T243 xl:.l - 9x l:! - 12xl - 4xo - 81x4 - 216xJ - 216x'/. - 96x - 16 T242 XL~ - 4Xll + 18x l:! - 4xti - 36x5 + 81x4 + 16xJ + 108x'/. + 16 T241 Xl2 + XlU - 3x l:! - Xo + 6x4 - 3 T240 x 1 :.1 + 4x l:! - 2xo + 4X4 - x:.l + 7 T239 x 1 :.1 - 12xll + 9x l:! - 32xti + 48xb - 18x4 - 64xJ + 144x'/. - 108x + 27 T23R x 1 :.1 - 2XlU - 3x l:! + 4xo + 2X4 + 4x'/. - 2 T237 Xi:.l - 2x lU - xl:! - 4xti + 2 T..23fL Xl'/. + XlU + Xll - XO + 4X4 + 1 T2::1F. Xl'/. - 2XlU - xl:! + 4X4 + 4x2 + 2 T2::14 Xi:.l - x!l - 3xJ + 4 T233 x 1 :.1 - 4x.1 - 6 T232 x 1 :.1 - 13x l:! - 26x( - llxti + 6x5 + 25x4 + 78x J + 114x:'! + 76x + 19 T231 Xi:.l + 36x l:! - 48x7 - 65xo + 162xo + 459x4 - 1488x.1 + 1512x :.l

-672x + 112


T230 Xl:.! + x lU - 3x!! + 4X4 + 1 T229 xl:.! - 24x9 + 108x!! - 720xo + 324x° + 2349x'i - 1728x :.i - 1296x2

+5832x + 5103 T228 Xl:.! - 6xlU - 24xll - 15x!! + 96x7 + 786xti - 912xo + 1974x4

-6992x3 + 16896x2 - 19728x + 12609 T227 Xl:.! - 3xlU - 3x!! - x b + 2 T226 XU + x lU - x!! - 2Xb + x:.! + 1 T22Fi XU - 3xlU + 2xo + 2x'i - 3 T224 XU - 2x lU + x!! + 3x4 - 2x:'! + 3 T22~ xl:.! ....., 3xlU - 5xb + 6x4 + 3x:.! - 3 T222 xl:.! - 4xti + 3x:.! - 1 T221 Xl:.! - 2XlU - x!! + 6xti - X4 - 4x2 - 1 T 220 Xl2 - 4x !l - 12x!! + 34xo - 12x o + 45x'i + 42x~ + 10 T219 Xl~ - Xo + 2x'i + x~ + 1 T21R Xl~ - 2XH + 22x !l - 88x( + 176xo - 176x;j + 64x + 4 T217 Xl~ - 6xo - 8X;1 - 4 T216 Xl:.! - 12x lU - 8Xll + 162x4 + 432x;1 + 432x:'! + 192x + 32 T215 Xl:.! - 12xlU - 12xll + 54x!! + 108x7 - 84xti - 324xo - 63x'i

+436x3 + 216x2 - 336x - 304 T214 XU - 12x !l + 18x!! - 56xb + 138x4 - 96x;1 + 72x:.! + 72 T21~ Xl:.! - 2x!l-1 T212 Xl:.! - 16x !l - 72x!! - 192x'( - 800xb - 1824x4 + 4608x:.! - 2048 T211 Xl:.! -135x!! -180x7 + 210xti + 540xo + 765x4 + 1160x;1 + 1080x:.!

+480x + 80 T210 Xl:.! - 4Xll + 8xti - 36xo + 105x4 -120x;1 + 90x:.! - 36x + 9 T 20Q Xl:.! - 8x9 + 18x!! - 24x7 + 24xti - 33x4 - 16x :.i - 48x - 8 T20R Xl:.! - 3xlU + 3xti + 3x4 + 3 T 207 Xl:.! - 6xlU - 8Xll + 9x!! - 60xb + 207x4 - 256x;j -1494x:'!

-1848x - 793 T206 Xl2 - 12x !l + 15x!! - 12x o + 18x4 - 64x;j + 96x2 - 36x + 9 T 20Fi Xl:.! - 208xb - 312xo - 117x4 - 832x;1 - 1872x:'! - 1404x - 351 T 204 xl:.! + 260xll + 63x!! - 648x7 - 780xb + 108xo + 2133x'i - 32x;J

-900x2 + 125 T20~ Xl:.! - x lU - X4 + x:.! + 1 T 202 Xl:.! - 2XlU - 4xb + 6x'i + 4x:.! + 4 T 201 Xl:.! - 3x lU + 3xb + 6x4 + 3x:.! + 3 Two Xl:.! + 6xlU + 9x!! - 8xti - 24x4 + 52 T199 Xl:.! - 2XlU - 4x8 - xti + x'i + 4 Ti98 xl:.! - 2XlU - x!! + 6x4 - 4x2 + 2 Tlil7 Xl:.! - 14x lU + 70x!! - 152x6 + 144x4 - 50x2 + 5 TIQf\ Xl:.! - 2x!! - 4xti + 6x4 + 4x~ - 1 TtQi'i xl:.! - x!! + 4xti - 5x4 + 4x:.! + 1 T194 Xl:.! - 4Xll - 32x9 + 198x!! + 216x7 + 1032xti - 384xo + 801x4

-452x3 + 32x2 + 4 T193 XU + 6xb + 6x4 + 3 T192 XU - 6xlU + x!! + 36xb - 30x4 - 28x:'! + 18 TIQl XU + x lU + 2x!! - x b + 2X4 - 3x:.! + 1 TIQO Xl:.! + 2XlU - 13x!! + 36xb + 15x4 - 38x:.! -19 TIRQ Xl:.! - 6xlU + 7x!! + 12xb - 16x4 - 8x:.! + 5


TIRR Xl~ - 2XlU + 5xo + 5x~ - 1 T1R7 Xl:.! - 6x lS - 2xo - 3x:.! + 1 T1Rn Xl:.! - 2XlS - 3x:.! - 1 T185 Xl~ - XIS - X4 + 2 T I84 Xl:.! - 2XlU - 2XlS + 4Xfi - X4 + 6x2 + 1 TI83 x12 - 6x lU + 49x8 - 2464xo + 388x4 + 80x2 + 4 TI82 Xl:.! - 8x9 + 6x" + 20xo - 24x" + 18x4 - 16x3 + 24x2 + 8 TI8I Xl:.! - 18x8 - 36xfi - 72x5 + 54x4 - 144x3 - 216x:.! - 72 TI80 x12 - 2XlU + 5x8 - 8Xfi + 6x4 - 4x2 + 1 L2(11) Xl:.! - 4Xll + 396x8 - 1056x7 + 2112xfi + 52272x4 - 69696x3

+278784x2 - 211968x + 2336832 T17R Xl2 - x'd + 4x.1 - 1 T177 XU - 4x'd + 4x.1 + 2 T17n Xl:.! + 4xo - 8x.1 + 8 T I75 XU - 12xlU - 8x'd + 36x lS + 48x'( - 65xo - 162x t> + 135x4 + 624x.1

+648x2 + 288x + 48 TI74 Xl:.! + 12x lU - 24x !J - 144x'( - 136xo + 432x t> + 1092x4 - 1440x~

-1152x -128 T~3 Xl:.! - 36x lS - 48x'( - 32xo + 162x4 - 288x:'! + 128 TI72 Xl:.! - 2Xll + 16x lU - 68x9 - 530x lS - 300x7 + 5380xo + 19304x t>

+27280x4 + 19880x3 + 10476x2 + 2704x + 676 T171 xL.l _ 8x !J - 36x lS - 72x t> + 81x4 + 64x.1 - 144x~ + 64 T 170 Xl:.! - x!J + 2xo + 4x3 + 3 TI69 xl:.! - 8X 3 + 18 T I6R Xl2 - lOxo - 12x3 - 2 TIn7 XU - 3x.1 + 3 Tlllll XU + 18xlU + 135x1! + 348xo + 63x4 - 512x3 - 270x~ + 729 T I6S XU - 16x 'd + 12x" + 256x3 - 576x:'! + 432x - 108 TI64 XU + 3xll - 6x lU - 33x 'd - 30x" + 54x7 + 155xfi + 180x5 + 192x4

+272x3 + 300x2 + 192x + 64 TIIl3 Xl2 _ x 8 - 2Xfi + X4 - 2X2 + 1 Tl1l2 Xl~ - 2x1! - 8xo + 14x4 - 16x~ + 4 TIll I Xl:.! - x" + 2xo + X4 + 2x~ + 1 Two XU + XlU + x" + Xo - 4X4 + 5 T 1S9 Xl2 + 4x lU - 4x" - 24xo - X4 + 32x:'! + 8 T15R Xl~ - x" - 2xo + 2x~ + 1 T157 Xl:.! - 8x 'd + 24xf + 44xo - 51x4 + 48x.1 - 72x~ + 16 Tl.~n XU - 6xo - 8x.1 - 1 T 155 Xl:.! - 2XlU - 3x1! + 2 T 154 Xl:.! + 10xlS - 4xo + 49x4 + 52x:'! + 104 TI53 XU - 3x lU + 5x lS - 8Xfi + lOx4 - 12x2 + 8 TI52 Xl:.! - 4x lS - 2xo + 4X4 - 1 TI5I Xl:.! - 3x lS - 2 T I50 Xl:.! - Xfi - 3x4 + 2x:'! + 2 T l49 Xl:.! - 9x4 - 6 TI4R Xl:.! - 2XlO + 2x" - 2xo - 2X4 - 2x:'! - 1 TI47 x12 - 9x4 - 12 T141l Xl2 - 2XlU - XIS - 2xo - 2X4 - 8x~ + 8 TI45 xl:.! + 6x" + 4xo - 18x4 - 24x~ - 8

Realizations over Q> for Transitive Groups 421

T144 Xl:.! + 6x lU + 4x~ - 24xO - 21x'l + 22x~ + 4 T 143 Xl~ - 6x lU + 24x~ - 56x tJ + 93x 'l - 90x~ + 51 T142 Xl2 + 3x~ + 4x tJ + 6x 'l + 3 T141 Xl~ + 3x~ - 3 T140 Xl~ + 4x'l - 4 T1::!C) Xl~ + x~ - 3xtJ + x'l + 1 Tl::!R Xl~ - x'l + 1 TU7 Xl:.! + x lU + xIS - x'l + X:.! - 1 T136 Xl:.! - 2XlU - 5x 'l - 2x:'! + 4 Tl35 Xl:.! - 18x~ - 24x o + 27x 'l + 36x~ - 6 Tl::!4 Xl~ - 7x lU + 14x~ - 21x'l + 7x~ + 7 Tl33 Xl~ - 12x lU - 8x \J + 54x~ + 72x7 - 84xli - 216xb - 63x 'l + 376x;j

+216x2 - 480x + 208 T 132 Xl2 - 4x lO - 36x' - 12x tJ + 144x b + 228x 'l + 208x;j + 360x~

+464x + 216 T131 Xl:.! - 3xlU - 2xY + 54xlS + 72x' + 402x6 + 756x5 + 5445x4

+13288x3 + 13176x2 + 5856x + 976 T l30 Xl:.! - xY + 5x6 - 8x3 + 4 T 129 Xl:.! - 2x lU - 12x :ol - 7x~ + 16x' + 50xo + 24x b - 46x4 - 40x3

+8x2 + 8x-l T 128 Xl:.! - 20xY + 81x8 - 272xti + 360xb + 756x4 - 192x3 + 144x:'!

+648x + 243 T127 Xl:.! - 54x :ol - 315x~ + 4372xo + 3996xb - 22005x 'l - 13176x3

+17484x2 - 22518x + 1775 T126 Xl:.! + XIS + xti - 2x'l - x:.! + 1 Tl25 Xl:.! - 2XlS - 2xti + x'l + 2x:'! - 1 T124 Xl2 - 2XlO - 5xlS + 35x 'l - 30x~ + 5 T123 Xl2 _ 2XlO + lOx6 - 8x:.! + 1 Tl22 Xl:.! - 2Xll - 3xlO - 6x9 + 21x~ - 32x7 + 37xti - 16xb + llx'l

+32x3 - x 2 + 20x + 1 T121 Xl:.! - x 9 + 2x3 + 1 T l20 Xl:.! - 2x9 - 6x3 + 9 T 119 xl:.! - 8xti - 8x3 - 2 TllR Xl~ + 8xo - 8x;J + 2 Tl17 Xl~ - 2x :ol + Xli + 5 TllG Xl~ - 2x \J + 4x;J + 4 Tllfi Xl~ - 2x i:S + 3x 'l - 4 T~ Xl~ - x'l - 1 TIl::! Xl~ - xi:S + 4 T112 Xl:.! - 3x i:S + 9x 'l + 9 Tll1 Xl:.! - 6x i:S + 68xli + 105x 'l + 36x:.! + 12 TllO Xl~ + xl! - Xli - x'l - 1 TID9 Xl:.! + x lU - 4x:.! + 1 TlO8 Xl:.! - 3xlS - 4xti + 6x 'l + 4 TlO7 Xl:.! + 6x lU + 3xlS - 28xli - 21x4 + 30x:.! + 5 TW6 Xl:.! + 3xlU - 2x~ - 9Xli + 5x~ + 1 T10fi Xl~ - 7x lU + 7x~ + 14xo - 16x'l - 5x~ + 5 TiQ4 Xl~ + 6x lU + 12x i:S + 8xtJ - 3x 'l - 6x:': - 1 T103 xl:.! - 2x i:S + 5xtJ - 2x'I + 1


T102 XU - 5XLU + 20x lS - 70xo + 145x 'l - 280x~ + 208 T101 XU - XIS + Xo - x'l + 1 TlOO Xl~ - XLU + XIS + 4xo - x'l - x~ - 1 T99 x12 + 12x lO + 60x8 + 160xfi + 228x4 + 144x2 + 8 T98 X12 + 214x lO - 1046693xlS + 18491564xfi - 29606993x4

+937950x2 + 8560357 T97 Xl~ + XIS + 9x 'l + 1 T96 X12 - 3x4 - 4 T95 XU - XLU + 3xo - 2x'l - 3x~ + 1 T94 X12 _ 57xlS - 38xD + 318x 'l - 204x2 + 17 T93 Xl:': + lOxlO + 28x8 + 6x6 - 43x4 + 6x:': + 3 Tq2 XU - 9x 'l - 9 Tql Xl:': + 5xlU + 9x lS + 8xo + 2x'l - 12x~ + 16 Tqn Xl:': + 2xlU - Xo + 2x:': + 1 TRQ x12 - 3x 'l + 1 TRR x12 - 6x lS - 4x0 - 3x 'l - 18x~ + 3 TR7 Xl:': + 6XLU + 9x lS - 4xo - 12x'l + 1 TR6 Xl:': + 2x lS - 2 T85 Xl:': - 3Xll - 3xlU + 15xll - 15xlS - 33x7 + 29xo + 15xb - 30x4

-128x3 - 30x2 + 198x + 48 TR4 x12 - 4Xll + 2x LU + 12xll - 20xlS + 16xo - 6x 'l - 8x;j + 4x~ + 8x + 4 TR~ x12 + 3xo - x;j + 3 TR2 Xl:': - 12xLU + 54x1S - 116x6 + 129x4 - 72x:': - 16 T81 Xl:': + X6 + 2 T80 Xl:': - 90xlS + 160xfi - 135x4 + 7200x2 - 80 T79 XU - 4xlS + 4xo + 5x 'l - 4x~ + 2 T78 XU _Xli +x;j + 1 T77 XU - 2x LU + XIS + 6xo - 6x 'l + 1 T76 Xl:': + 2XlS - 2x6 + 5x 'l - 6x" + 1 T'l5 Xl:': - 2XlO - 2XlS + 6x6 + X4 - 6x2 + 1 T74 Xl:': - XlO + 2XlS + 4x6 - 3x 'l - 3x:': + 1 T73 XU + 12x LU - 3Xll + 54x lS - 27x'( + 122xo - 81xb + 165x 'l - 93x;j

+126x2 - 36x + 31 T72 Xl:': + 12x lO - 40x9 + 414x lS - 1416x( + 3388xo - 6552x1> + 800lx'l

-7448x3 + 7056x2 + 4704x + 2744 T71 Xl~ - 4Xll + 4xo + 3 T 7n Xl:': + 9xb - 18x;j + 9 T6Q Xl:': - 3xlO - 2XlS + 9Xfi - 5x2 + 1 Ts8 Xl:': + XlO + 6xlS + 3xo + 6x 'l + X~ + 1 T67 X12 _ XIS - Xo - X4 + 1 T66 x12 + 6x LU + 12xlS + 8xo - 3 T65 XU - 3x 'l +4 T64 Xl~ - XIS + 9x 'l - 1 T63 Xl~ - 6xLU + 104x0 + 93x4 + 18x2 + 4 T62 Xl~ + 6xLU - 104xb + 93x4 - 18x2 + 4 T61 x12 - 3x4 -1 T60 Xl~ -14x lS -7x 'l +4 T59 XU - 6x LU + 6x lS - 4xo - 3x4 + 3 T58 XU - 12xlS - 14x b + 9x4 + 12x:': + 1


T,,'7 Xl2 - 69x lO - 2091x8 - 7571:.&0 + 134691x4 + 960267x" + 1545049 T"", Xl2 _ 2x lO + x6 - 2x-z+ 1 T",,,, xU- - 30x lO + 348xlr - 1960x6 + 5505x4 - 7050x2 + 3025 Tt;4 x-U- - 6x8 + 9x4 + 12 T"


Tn Xl:t + lOxb + 5 llo Xl:t + 16 T9 Xl:.! + 3xlS + 4xb + 3x4 + 1 Ts Xl:.! - 6xlU - 8x9 + 9x8 + 12x7 - 20x6 + 9x4 - 24x3 - 4 T7 Xl:.! + 4x lU - xIS - x'l + 4x :t + 1 Til Xl~ + 2X lU - 6x e! + 2X b - 6x'l + 2x~ + 1 T5 Xl~ - 80x lU + 1820xe! - 13680xb + 29860x'l - 2720x:' + 32 T4 Xl:.! + 6xlS + 26xo - 63x 'l + 162x:' + 81 T.1 Xl:.! + 36 T2 Xl:.! - x 6 + 1 Tl Xl:.! - Xll + x lU - Xli + xes - Xf + X O - xt> + X4 - x 3 + x:t - X + 1

References

Abhyankar, S.S. (1957): Coverings of algebraic curves. Amer. J. Math. 79, 825-856 Artin, E. (1925): Theorie der Zopfe. Abh. Math. Sem. Univ. Hamburg 4, 47-72 Artin, E. (1947): Theory of braids. Ann. of Math. 48, 101-126 Artin, E. (1967): Algebraic numbers and algebraic functions. Gordon and Breach,

New York Aschbacher, M. (1986): Finite group theory. Cambridge University Press, Cam-

bridge Aschbacher, M. et al. (eds.) (1984): Proceedings of the Rutgers group theory year,

1983-1984. Cambridge University Press, Cambridge Auslander, M., Brumer, A. (1968): Brauer groups of discrete valuation rings. Indag.

Math. 30, 286-296 Bayer-Fluckiger, E. (1994): Galois cohomology and the trace form. Jahresber.

Deutsch. Math.-Verein. 96, 35-55 Beckmann, S. (1989): Ramified primes in the field of moduli of branched coverings

of curves. J. Algebra 125, 236-255 Beckmann, S. (1991): On extensions of number fields obtained by specializing

branched coverings. J. reine angew. Math. 419, 27-53 Belyi, G.V. (1979): On Galois extensions of a maximal cyclotomic field. Izv. Akad.

Nauk SSSR Ser. Mat. 43, 267-276 (Russian) [English transl.: Math. USSR Izv. 14 (1979),247-2561

Belyi, G.V. (1983): On extensions of the maximal cyclotomic field having a given classical Galois group. J. reine angew. Math. 341, 147-156

Benson, D.J. (1991): Representations and cohomology I. Cambridge University Press, Cambridge

Beynon, W.M., Spaltenstein, N. (1984): Green functions of finite Chevalley groups of type En (n = 6,7,8). J. Algebra 88, 584--£14

Birman, J.S. (1975): Braids, links and mapping class groups. Princeton University Press, Princeton

Borel, A. (1991): Linear algebraic groups. Springer, New York Borel, A. et al. (1970): Seminar on algebraic groups and related finite groups. LNM

131, Springer, Berlin Bosch, S., Giintzer, U., Remmert, R. (1984): Non-archimedean analysis. Springer,

Berlin Heidelberg New York Brieskorn, E., Saito, K. (1972): Artin-Gruppen und Coxeter-Gruppen. Invent.

Math. 17, 245-271 Bruen, A.A., Jensen, C.U., Yui, N. (1986): Polynomials with Frobenius groups of

prime degree as Galois groups. J. Number Theory 24,305-359 Butler, G., McKay, J. (1983): The transitive groups of degree up to eleven. Comm.

Algebra 11, 863-911 Cartan, H., Eilenberg, S. (1956): Homological algebra. Princeton University Press,

Princeton

426 References

Carter, RW. (1985): Finite groups of Lie type: Conjugacy classes and complex characters. John Wiley and Sons, Chichester

Carter, RW. (1989): Simple groups of Lie type. John Wiley and Sons, Chichester Chang, B., Ree, R (1974): The characters of G2(q). Pp. 395-413 in: Symposia

Mathematica XIII, London Chow, W.-L. (1948): On the algebraic braid group. Ann. of Math. 49, 654-658 Cohen, A.M. et al. (1992): The local maximal subgroups of exceptional groups of

Lie type, finite and algebraic. Proc. London Math. Soc. 64, 21-48 Cohen, D.B. (1974): The Hurwitz monodromy group. J. Algebra 32,501-517 Conway, J.H. et al. (1985): Atlas of finite groups. Clarendon Press, Oxford Cooperstein, B.N. (1981): Maximal subgroups of G2(2n ). J. Algebra 70, 23-36 Crespo, T. (1989): Explicit construction of An type fields. J. Algebra 127, 452-461 Debes, P., Fried, M.D. (1990): Arithmetic variation of fibers in families of covers I.

J. reine angew. Math. 404, 106-137 Deligne, P., Lusztig, G. (1976): Representations ofreductive groups over finite fields.

Ann. of Math. 103, 103-161 Delone, B.N., Faddeev, D.K. (1944): Investigations in the geometry of the Galois

theory. Math. Sb. 15(57), 243-284 (Russian with English summary) Demuskin, S.P., Safarevic, I.R. (1959): The embedding problem for local fields. Izv.

Akad. Nauk. Ser. Mat. 23, 823-840 (Russian) [English transl.: Amer. Math. Soc. Transl. II 27, 267-288 (1963)1

Dentzer, R. (1989): Projektive symplektische Gruppen PSp4{p) als Galoisgruppen iiber Q{t). Arch. Math. 53, 337-346

Dentzer, R. (1995a): Polynomials with cyclic Galois group. Comm. Algebra 23, 1593-1603

Dentzer, R. (1995b): On geometric embedding problems and semiabelian groups. Manuscripta Math. 86, 199-216

Deriziotis, D.I. (1983): The centralizers of semisimple elements of the Chevalley groups E7 and Ea. Tokyo J. Math. 6, 191-216

Deriziotis, D.L, Michler, G.O. (1987): Character table and blocks of finite simple triality groups =b4 {q). Trans. Amer. Math. Soc. 303,39-70

Dew, E. (1992): Fields of moduli of arithmetic Galois groups. PhD-thesis, University of Pennsylvania

Digne, F., Michel, J. (1991): Representations of finite groups of Lie type. LMS Student Texts 21, Cambridge University Press, Cambridge

Digne, F., Michel, J. (1994): Groupes roouctifs non connexes. Ann. scient. Ec. Norm. Sup. 27, 345-406

van den Dries, L., Ribenboim, P. (1979): Application de la theorie des modeles aux groupes de Galois de corps de fonctions. C. R Acad. Sci. Paris 288, A789-A792

Douady, A. (1964): Determination d'un groupe de Galois. C. R. Acad. Sci. Paris 258, 5305-5308

Enomoto, H. (1976): The characters of the finite Chevalley groups G2 {q), q = 3f . Japan. J. Math. 2, 191-248

Enomoto, H., Yamada, H. (1986): The characters of G2(2n ). Japan. J. Math. 12, 325-377

Faddeev, D.K. (1951): Simple algebras over a field of algebraic functions of one variable. Trudy Mat. Inst. Steklov 38, 19(}-243 (Russian) [English transl.: Amer. Math. Soc. Transl. II 3, 15-38 (1956)1

Fadell, E. {1962}: Homotopy groups of configuration spaces and the string problem of Dirac. Duke Math. J. 29, 231-242

Fadell, E., Van Buskirk, J. (1962): The braid groups of E2 and 8 2 . Duke Math. J. 29,243-257

References 427

Feit, W. (1984): Rigidity of Aut(PSL2(p2», p == ±2 (mod 5), p =f 2. Pp. 351-356 in: M. Aschbacher et al. (eds.) (1984)

Feit, W. (1989): Some finite groups with nontrivial centers which are Galois groups. Pp. 87-109 in: Group Theory, Proceedings of the 1987 Singapore Conference. W. de Gruyter, Berlin-New York

Feit, W., Fong, P. (1984): Rational rigidity of G2(p} for any prime p > 5. Pp. 323-326 in: M. Aschbacher et al. (eds.) (1984)

Fleischmann, P. , Janiszczak, I. (1993): The semisimple conjugacy classes of finite groups of Lie type E6 and E7 • Comm. Algebra 21, 93-161

Folkers, M. (1995): Lineare Gruppen ungerader Dimension als Galoisgruppen iiber Q. IWR-Preprint 95-41, Universitat Heidelberg

Forster, O. (1981): Lectures on Riemann surfaces. Springer, New York Franz, W. (1931): Untersuchungen zum Hilbertschen Irreduzibilitatssatz. Math. Z.

33,275-293 Fresnel, J., van der Put, M. (1981): Geometrie analytique rigide et applications.

Birkhauser, Basel Fried, M.D. (1977): Fields of definition of function fields and Hurwitz families -

Groups as Galois groups. Comm. Algebra 5, 17-82 Fried, M.D. (1984): On reduction of the inverse Galois problem to simple groups.

Pp. 289-301 in: M. Aschbacher et al. (eds.) (1984) Fried, M.D., Biggers, R. (1982): Moduli spaces of covers and the Hurwitz mon-

odromy group. J. reine angew. Math. 335,87-121 Fried, M.D., Debes, P. (1990): Rigidity and real residue class fields. Acta Arith-

metica 56,291-323 Fried, M.D., Jarden, M. (1986): Field arithmetic. Springer, Berlin Fried, M.D., Volklein, H. (1991): The inverse Galois problem and rational points

on moduli spaces. Math. Ann. 290, 771-800 Fried, M.D., Volklein, H. (1992): The embedding problem over a Hilbertian PAC-

field. Ann. of Math. 135, 469-481 Fried, M.D. et al. (eds.) (1995): Recent developments in the inverse Galois problem.

Contemporary Math. vol. 186, AMS, Providence Frohlich, A. (1985): Orthogonal representations of Galois groups, Stiefel-Whitney

classes and Hasse-Witt invariants. J. reine angew. Math. 360,84-123 Geyer, W.-D., Jarden, M. (1998): Bounded realization of i-groups over global fields.

Nagoya Math. J. 150, 13-62 Gilette, R., Van Buskirk, J. (1968): The word problem and consequences for the

braid groups and mapping class groups of the 2-sphere. Trans. Amer. Math. Soc. 131, 277-296

Goss, D. (1996): Basic structures of function field arithmetic. Springer, Berlin Granboulan, L. (1996): Construction d'une extension reguliere de Q(T) de groupe

de Galois M24 • Experimental Math. 5,3-14 Grothendieck, A. (1971): Revetements etales et groupe fondamental. Springer,

Berlin Guralnick, R.M., Thompson, J.G. (1990): Finite groups of genus zero. J. Algebra

131, 303-341 Hafner, F. (1992): Einige orthogonale und symplektische Gruppen als Galoisgrup-

pen iiber Q. Math. Ann. 292, 587-618 Hansen, V.L. (1989): Braids and coverings. Cambridge University Press, Cambridge Haran, D., Volklein, H. (1996): Galois groups over complete valued fields. Israel J.

Math. 93, 9-27 Harbater, D. (1984): Mock covers and Galois extensions. J. Algebra 91, 281-293 Harbater, D. (1987): Galois coverings of the arithmetic line. In: Chudnovsky, D.V.

et al. (eds.): Number Theory, New York 1984-1985. Springer, Berlin

428 References

Harbater, D. (1994a): Abhyankar's conjecture on Galois groups over curves. Invent. Math. 117, 1-25

Harbater, D. (1994b): Galois groups with prescribed ramification. Pp. 35-60 in: N. Childress and J. Jones (eds.): Arithmetic Geometry. Contemporary Math. vol. 174, AMS, Providence

Harbater, D. (1995a): Fundamental groups and embedding problems in character-istic p. Pp. 353-369 in: M. Fried et al. (eds.) (1995)

Harbater, D. (1995b): Fundamental groups of curves in characteristic p. Pp. 656-666 in: Proc. Int. Congress of Math., Ziirich 1994 , Birkhauser

Hartshorne, R. (1977): Algebraic geometry. Springer, Berlin Heidelberg New York Hasse, H. (1948): Existenz und Mannigfaltigkeit abelscher Algebren mit vorgege-

bener Galoisgruppe iiber einem Teilkorper des Grundkorpers I. Math. Nachr. 1, 40-61

Hasse, H. (1980): Number theory. Springer, Berlin Hecke, E. (1935): Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe

durch algebraische Eigenschaften. Math. Ann. 111, 293-301 Higman, D., McLaughlin, J. (1965): Rank 3 subgroups of finite symplectic and

unitary groups. J. reine angew. Math. 218, 174-189 Hilbert, D. (1892): tiber die Irreducibilitat ganzer rationaler Functionen mit ganz-

zahligen Coeffizienten. J. reine angew. Math. 110, 104-129 Hilton, P.J., Stammbach, U. (1971): A course in homological algebra. Springer, New

York Hoechsmann, K. (1968): Zum Einbettungsproblem. J. reine angew. Math. 229,

81-106 Hoyden-Siedersleben, G. (1985): Realisierung der Jankogruppen J 1 und J 2 als Ga-

loisgruppen iiber Q. J. Algebra 97, 14-22 Hoyden-Siedersleben, G., Matzat, B.H. (1986): Realisierung sporadischer einfacher

Gruppen als Galoisgruppen iiber Kreisteilungskorpern. J. Algebra 101, 273-285 Hunt, D.C. (1986): Rational rigidity and the sporadic groups. J. Algebra 99, 577-

592 Huppert, B. (1967): Endliche Gruppen I. Springer, Berlin Huppert, B., Blackburn, N. (1982): Finite groups II. Grundlehren 242, Springer,

Berlin Hurwitz, A. (1891): tiber Riemann'sche Flachen mit gegebenen Verzweigungspunk-

ten. Math. Ann. 39, 1--{)1 Ihara, Y. (1991): Braids, Galois groups, and some arithmetic functions. Pp. 99-120

in: Proc. Int. Congress of Math., Kyoto 1990, Springer Ihara, Y. et a1. (eds.) (1989): Galois groups over Q. Springer, New York Ikeda, M. (1960): Zur Existenz eigentlicher galoisscher Korper beim Einbettungs-

problem flir galoissche Algebren. Abh. Math. Sem. Univ. Hamburg 24, 126-131 Inaba, E. (1944): tiber den Hilbertschen Irreduzibilitatssatz. Japan. J. Math. 19,

1-25 Ishkhanov, V.V., Lure, B.B., Faddeev, D.K. (1997): The embedding problem in

Galois theory. AMS, Providence Iwasawa, K. (1953): On solvable extensions of algebraic number fields. Ann. of

Math. 58, 548-572 . Jacobson, N. (1980): Basic algebra II. W. H. Freeman, New York Kantor W.M. (1979): Subgroups of classical groups generated by long root elements.

Trans. Amer. Math. Soc. 248, 347-379 Kleidman, P.B. (1987): The maximal subgroups of the finite 8-dimensional orthog-

onal groups pn;(q) and of their automorphism groups. J. Algebra 110, 173-242 Kleidman, P.B. (1988a): The maximal subgroups of the Steinberg triality groups

1)4(q) and of their automorphism groups. J. Algebra 115, 182-199

References 429

Kleidman, P.B. (1988b): The maximal subgroups of the Chevalley groups G2 (q) with q odd, the Ree groups ~2 (q) and of their automorphism groups. J. Algebra 117,30-71

Kleidman, P.B., Liebeck, M. (1990): The subgroup structure of the finite classical groups. LMS lecture notes 129, Cambridge University Press, Cambridge

Kleidman, P.B., Parker, R.A., Wilson, R.A. (1989): The maximal subgroups of the Fischer group Fi23 . J. London Math. Soc. (2) 39,89-101

Kleidman, P.B., Wilson, RA. (1987): The maximal subgroups of Fh2. Math. Proc. Cambridge Philos. Soc. 102, 17-23

Kleidman, P.B., Wilson, RA. (1988): The maximal subgroups of J4. Proc. London Math. Soc. 56,484-510

Klein, F. (1884): Vorlesungen iiber das Ikosaeder. Teubner, Leipzig Klein, F., Fricke, R (1890): Vorlesungen iiber die Theorie der elliptischen Modul-

funktionen I. Teubner, Leipzig Kliiners, J., Malle, G. (1998): Explicit Galois realization of transitive groups of

degree up to 15. IWR-Preprint 99-13 Kochendarffer, R (1953): Zwei Reduktionssiitze zum Einbettungsproblem fiir abel-

sche Algebren. Math. Nachr. 10, 75-84 Kapf, U. (1974): tiber eigentliche Familien algebraischer Varietiiten iiber affinoiden

Riiumen. Schriftenreihe Math. Inst. Univ. Miinster, 2. Serie, Heft 7. Krull, W., Neukirch, J. (1971): Die Struktur der absoluten Galoisgruppe iiber dem

Karper ]R(t). Math. Ann. 193, 197-209 Kucera, J. (1994): tiber die Brauergruppe von Laurentreihen- und rationalen Funk-

tionenkarpern und deren DualiUit mit K-Gruppen. Dissertation, Universitiit Hei-delberg

Kuyk, W. (1970): Extensions de corps Hilbertiens. J. Algebra 14, 112-124 Landazuri, V., Seitz, G.M. (1974): On the minimal degrees of projective represen-

tations of the finite Chevalley groups. J. Algebra 32, 418-443 Lang, S. (1982): Introduction to algebraic and abelian functions. Springer, New

York Linton, S.A. (1989): The maximal subgroups of the Thompson group. J. London

Math. Soc. (2) 39, 79-88 [correction: ibid. 43 (1991), 253-254] Linton, S.A., Wilson, R.A. (1991): The maximal subgroups of the Fischer groups

Fh4 and Fi~4' Proc. London Math. Soc. (3) 63, 113-164 Liu, Q. (1995): Tout groupe fini est un groupe de Galois sur Qp(t), d'apres Harbater.

Pp. 261-265 in: M. Fried et al (eds.) (1995) Liibeck, F., Malle, G. (1998): (2, 3}-generation of exceptional groups. To appear in

J. London Math Soc. Lusztig, G. (1977): Representations of finite Chevalley groups. CBMS Regional

Conference Series in Mathematics 39 Lusztig, G. (1980): On the unipotent characters of the exceptional groups over finite

fields. Invent. Math. 60, 173-192 Lusztig, G. (1984): Characters of reductive groups over a finite field. Annals of

Math. Studies 107, Princeton University Press, Princeton Lyndon, R.C., Schupp, P.E. (1977): Combinatorial group theory. Springer, Berlin Madan, M., Rzedowski-Calderon, M., Villa-Salvador, G. (1996): Galois extensions

with bounded ramification in characteristic p. Manuscripta Math. 90, 121-135 Malle, G. (1986): Exzeptionelle Gruppen vom Lie-Typ als Galoisgruppen. Disser-

tation, Universitiit Karlsruhe Malle, G. (1987): Polynomials for primitive nonsolvable permutation groups of de-

gree d ~ 15. J. Symb. Compo 4, 83-92 Malle, G. (1988a): Polynomials with Galois groups Aut(M22), M22 , and PSL3(lF4).2

over Q. Math. Compo 51, 761-768

430 References

Malle, G. (1988b): Exceptional groups of Lie type as Galois groups. J. reine angew. Math. 392, 70-109

Malle, G. (1991): Genus zero translates of three point ramified Galois extensions. Manuscripta Math. 71,97-111

Malle, G. (1992): Disconnected groups of Lie type as Galois groups. J. reine angew. Math. 429, 161-182

Malle, G. (1993a): Polynome mit Galoisgruppen PGL2(p) und PSL2(p) iiber Q(t). Comm. Algebra 21, 511-526

Malle, G. (1993b): Generalized Deligne-Lusztig characters. J. Algebra 159, 64-97 Malle, G. (1993c): Green functions for groups of types E6 and F4 in characteristic 2.

Comm. Algebra 21, 747-798 Malle, G. (1996): GAR-Realisierungen klassischer Gruppen. Math. Ann. 304, 581-

612 Malle, G., Matzat, B. H. (1985): Realisierung von Gruppen PSL2(lFp) als Galois-

gruppen iiber Q. Math. Ann. 272, 549-565 Malle, G., Sonn, J. (1996): Covering groups of almost simple groups as Galois groups

over Qab(t}. Israel J. Math. 96,431-444. Matsumura, H. (1980): Commutative algebra. Benjamin/Cummings, Reading Matzat, B.H. (1979): Konstruktion von Zahlkorpern mit der Galoisgruppe Mll iiber

Q( .;=IT). Manuscripta Math. 27, 103-111 Matzat, B.H. (1984): Konstruktion von Zahl- und Funktionenkorpern mit vor-

gegebener Galoisgruppe. J. reine angew. Math. 349, 179-220 Matzat, B.H. (1985a): Zwei Aspekte konstruktiver Galoistheorie. J. Algebra 96,

499-531 Matzat, B.H. (1985b): Zum Einbettungsproblem der algebraischen Zahlentheorie

mit nicht abelschem Kern. Invent. Math. 80, 365-374 Matzat, B.H. (1986): Topologische Automorphismen in der konstruktiven Galois-

theorie. J. reine angew. Math. 371, 16-45 Matzat, B.H. (1987): Konstruktive Galoistheorie. LNM 1284, Springer, Berlin Matzat, B.H. (1989): Rationality criteria for Galois extensions. Pp. 361-383 in: Y.

Ihara et al. (eds.) (1989) Matzat, B.H. (1991a): Zopfe und Galoissche Gruppen. J. reine angew. Math. 420,

99-159 Matzat, B.H. (1991b): Frattini-Einbettungsprobleme iiber Hilbertkorpern. Manu-

scripta Math. 70, 429-439 Matzat, B.H. (1991c): Der Kenntnisstand in der konstruktiven Galoisschen Theorie.

Pp. 65-98 in G.O. Michler and C.M. Ringel (eds.) (1991) Matzat, B.H. (1992): Zopfgruppen und Einbettungsprobleme. Manuscripta Math.

14,217-227 Matzat, B.H. (1993): Braids and decomposition groups. Pp. 179-189 in: S. David

(ed.): Seminaire de theorie des nombres, Paris 1991-1992. Birkhauser, Boston Matzat, B.H. (1995): Parametric solutions of embedding problems. Pp. 33-50 in:

M. Fried et al. (eds.) (1995) Matzat, B.H., Zeh-Marschke, A. (1986): Realisierung der Mathieugruppen Mll und

M12 als Galoisgruppen iiber Q. J. Number Theory 23, 195-202 McLaughlin, J. (1969): Some subgroups of SLn(lFp}. Ill. J. Math. 13, 108-115 Melnicov, O.V. (1980): Projective limits of free profinite groups. Dokl. Akad. Nauk

SSSR 24, 968-970 Mestre, J.-F. (1990): Extensions regulieres de Q(T) de groupe de Galois An. J.

Algebra 131, 483-495 Mestre, J.-F. (1994): Construction d'extensions regulieres de Q(t) it. groupes de

Galois SL2(H) et I\h2. C. R. Acad. Sci. 319, 781-782

References 431

Mestre, J.-F. (1995): Construction polynomiales et theorie de Galois. Pp. 318-323 in: Proc. Int. Congress of Math., Ziirich 1994 , Birkhauser

Michler, G.O., Ringel, C.M. (eds.) (1991): Representation theory of finite groups and finite-dimensional algebras. Birkhauser, Basel

Mizuno, K. (1977): The conjugate classes of Chevalley groups of type E6 • J. Fac. Sci. Univ. Tokyo 24, 525-563

Mizuno, K. (1980): The conjugate classes of unipotent elements of the Chevalley groups E7 and E8 . Tokyo J. Math. 3, 391-461

Nagata, M. (1977): Field theory. Marcel Dekker, New York Nakamura, H. (1997): Galois rigidity of profinite fundamental groups. Sugaku Ex-

positiones 10, 195-215 Narkiewicz, W. (1990): Elementary and analytic theory of algebraic numbers.

Springer, Berlin Nobusawa, N. (1961): On the embedding problem of fields and Galois algebras.

Abh. Math. Sem. Univ. Hamburg 26,89-92 Noether, E. (1918): Gleichungen mit vorgeschriebener Gruppe. Math. Ann. 78,

221-229 Norton, S.P., Wilson, R.A. (1989): The maximal subgroups of F 4(2n) and its auto-

morphism group. Comm. Algebra 17, 2809-2824 Pahlings, H. (1988): Some sporadic groups as Galois groups. Rend. Sem. Math.

Univ. Padova 79, 97-107 Pahlings, H. (1989): Some sporadic groups as Galois groups II. Rend. Sem. Math.

Univ. Padova 82, 163-171 [correction: ibid. 85 (1991),309-310] Pop, F. (1994): ~ Riemann existence theorem with Galois action. Pp. 193-218 in:

G. Frey and J. Ritter (eds.): Algebra and number theory. W. de Gruyter, Berlin Pop, F. (1995): Etale Galois covers of affine smooth curves. Invent. Math. 120,

555-578 Pop, F. (1996): Embedding problems over large fields. Ann. of Math. 144, 1-34 Popp, H. (1970): Fundamentalgruppen algebraischer Mannigfaltigkeiten. Springer,

Berlin Porsch, U. (1993): Greenfunktionen der endlichen Gruppen E6(q), q = 3n . Diplom-

arbeit, Universitat Heidelberg Przywara, B. (1991): Zopfbahnen und Galoisgruppen. IWR-Preprint 91-01, Univer-

sitat Heidelberg Raynaud, M. (1994): Revetements de la droite affine en caracteristique p > 0 et

conjecture d'Abhyankar. Invent. Math. 116,425-462 Reichardt, H. (1937): Konstruktion von Zahlkorpern mit gegebener Galoisgruppe

von Primzahlpotenzordnung. J. reine angew. Math. 177, 1-5 Reiter, S. (1997): GAR-Realisierungen klassischer Gruppen YOm Lie-Typ. Disser-

tation, Universitat Heidelberg Ribes, L. (1970): Introduction to profinite groups and Galois cohomology. Queen's

University, Kingston Rzedowski-Calderon, M. (1989): Construction of global function fields with nilpo-

tent automorphism groups. Bol. Soc. Mat. Mexicana 34, 1-10 Safarevic, LR. (1947): On p-extensions. Math. Sb. Nov. Ser. 20 (62), 351-363 (Rus-

sian) [English transl.: Amer. Math. Soc. Transl. II 4,59-72 (1956)] Safarevic, LR. (1954a): On the construction of fields with a given Galois group of

order f.a. Izv. Akad. Nauk. SSSR 18, 216-296 (Russian) [English transl.: Amer. Math. Soc. Transl. II 4, 107-142 (1956)]

Safarevic, LR. (1954b): On an existence theorem in the theory of algebraic numbers. Izv. Akad. Nauk. SSSR 18, 327-334 (Russian) [English transl.: Amer. Math. Soc. Transl. II 4, 143-150 (1956)]

432 References

Safarevic, I.R. (1954c): On the problem of imbedding fields. Izv. Akad. Nauk. SSSR 18, 389-418 (Russian) [English transl.: Amer. Math. Soc. Transl. II 4, 151-183 (1956)]

Safarevic, I.R. (1954d): Construction of fields of algebraic numbers with given solv-able Galois group. Izv. Akad. Nauk. SSSR 18, 525--578 (Russian) [English transl.: Amer. Math. Soc. Transl. II 4, 185--237 (1956)]

Safarevic, I.R. (1958): The embedding problem for split extensions. Dokl. Akad. Nauk SSSR 120, 1217-1219 (Russian)

Safarevic, I.R. (1989): Factors of descending central series. Math. Notes 45,262-264 Saltman, D.J. (1982): Generic Galois extensions and problems in field theory. Adv.

Math. 43, 250--283 Saltman, D.J. (1984): Noether's problem over algebraically closed fields. Invent.

Math. 77, 71-84 Scharlau, W. (1969): tiber die Brauer-Gruppe eines algebraischen Funktionenkor-

pers einer Variablen. J. reine angew. Math. 239/240, 1--6 Scharlau, W. (1985): Quadratic and Hermitean forms. Springer, Berlin Schmidt, U. (1998): Galoisrealisierungen exzeptioneller Gruppen in Charakteris-

tik 2, in preparation Schneps, L. (1992): Explicit construction of extensions of Q{t) of Galois group An

for n odd. J. Algebra 146,117-123 Scholz, A. (1929): tiber die Bildung algebraischer Zahlkorper mit auflosbarer galois-

scher Gruppe. Math. Z. 30, 332-356 Scholz, A. (1937): Konstruktion algebraischer Zahlkorper mit beliebiger Gruppe

von Primzahlpotenzordnung I. Math. Z. 42, 161-188 Seidelmann, F. (1918): Die Gesamtheit der kubischen und biquadratischen Gle-

ichungen mit Affekt bei beliebigem Rationalitatsbereich. Math.Ann 78, 230--233 Seifert, H., Threllfall, W. (1934): Lehrbuch der Topologie. Teubner, Leipzig Serre, J.-P. (1956): Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier

6, 1-42 Serre, J.-P. (1959): Groupes algebriques et corps de classes. Hermann, Paris Serre, J.-P. (1964): Cohomologie galoisienne. Springer, Berlin Serre, J.-P. (1979): Local fields. Springer, New York Serre, J.-P. (1984): L'invariant de Witt de la forme Tr(x2 ). Comment. Math. Hel-

vetici 59, 651--679 Serre, J.-P. (1988): Groupes de Galois sur Q. Asterisque 161-162, 73-85 Serre, J.-P. (1990): Construction de revetements etales de la droite affine en car-

acteristique p. C. R. Acad. Sci. Paris 311, 341-346 Serre, J.-P. (1992): Topics in Galois theory. Jones and Bartlett, Boston Shabat, G.B., Voevodsky, V.A. (1990): Drawing curves over number fields. Pp.

199-227 in: P. Cartier et al. (eds.): The Grothendieck Festschrift III. Birkhauser, Boston

Shatz, S.S. (1972): Profinite groups, arithmetic and geometry. Princeton University Press, Princeton

Shih, K.-y. (1974): On the construction of Galois extensions of function fields and number fields. Math. Ann. 207, 99-120

Shih, K.-y. (1978): p-division points on certain elliptic curves. Compositio Math. 36,113-129

Shinoda, K. (1974): The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic 2. J. Fac. Sci. Univ. Tokyo 21, 133-159

Shoji, T. (1974): The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic p t= 2. J. Fac. Sci. Univ. Tokyo 21,1-17

Shoji, T. (1982): On the Green polynomials ofa Chevalley group oftype F4. Comm. Algebra 10, 505--543

References 433

Silverman, J.H. (1986): The arithmetic of elliptic curves. Springer, New York Simpson, W., Frame, J. (1973): The character tables for SL3(q), SU3(q), PSL3(q),

U3 (q). Can. J. Math. 25, 486-494 Smith, G.W. (1991): Generic cyclic polynomials of odd degree. Comm. Algebra 19,

3367-3391 Smith, G.W. (1993); Some polynomials over Q(t) and their Galois groups. To appear

in Math. Compo Sonn, J. (1990): On Brauer groups and embedding problems over function fields.

J. Algebra 131, 631-640 Sonn, J. (1991): Central extensions of Sn as Galois groups of regular extensions of

Q(T). J. Algebra 140, 355-359 Sonn, J. (1994a); Brauer groups, embedding problems, and nilpotent groups as

Galois groups. Israel J. Math. 85,391-405 Sonn, J. (1994b): Rigidity and embedding problems over Q3b(t). J. Number Theory

47,398-404 Spaltenstein, N. (1982); Caracteres unipotents de 3D4(lFq ). Comment. Math. Hel-

vetici 57, 676-691 Steinberg, R. (1967); Lectures on Chevalley groups. Yale University Steinberg, R. (1968): Endomorphisms oflinear algebraic groups. Mem. Amer. Math.

Soc. 80, AMS, Providence Stocker, R., Zieschang, H. (1988): Algebraische Topologie. Teubner, Stuttgart Stoll, M. (1995): Construction of semiabelian Galois extensions. Glasgow Math. J.

37,99-104 Suzuki, M. (1982): Group theory. Springer, Berlin Swallow, J.R. (1994): On constructing fields corresponding to the An's of Mestre

for odd n. Proc. Amer. Math. Soc. 122,85-90 Swan, R.G. (1969): Invariant rational functions and a problem of Steenrod. Invent.

Math. 7, 148-158 Tate, J. (1962); Duality theorems in Galois cohomology. Pp. 288-295 in: Proc. Int.

Congress of Math., Stockholm Tate, J. (1966): The cohomology groups of tori in finite Galois extensions of number

fields. Nagoya Math. J. 27, 709-719 Thompson, J.G. (1973): Isomorphisms induced by automorphisms. J. Austral.

Math. Soc. 16, 16-17 Thompson, J.G. (1984a): Some finite groups which appear as Gal(L/ K) where

K $ Q(J.£n). J. Algebra 89,437-499 Thompson, J.G. (1984b): Primitive roots and rigidity. Pp. 327-350 in: M. As-

chbacher et al. (eds.) (1984) Thompson, J.G. (1986): Regular Galois extensions of Q(x), Pp. 210-220 in: Than

Hsio-Fu (ed.): Group theory, Beijing 1984. Springer, Berlin Tschebotarow, N.G., Schwerdtfeger, H. (1950): Grundziige der Galoisschen Theorie.

Noordhoff, Groningen Uchida, K. (1980); Separably Hilbertian fields. Kodai Math. J. 3, 83-95 Vila, N. (1985): On central extensions of An as Galois groups over Q. Arch. Math.

44,424-437 Volklein, H. (1992a): Central extensions as Galois groups. J. Algebra 146,144-152 Volklein, H. (1992b): GLn(q) as Galois group over the rationals. Math. Ann. 293,

163-176 Volklein, H. (1993); Braid group action via GLn(q) and Un(q) and Galois realiza-

tions. Israel J. Math. 82, 405-427 Volklein, H. (1994): Braid group action, embedding problems and the groups

PGLn(q) and pUn(q2). Forum Math. 6, 513-535

434 References

Volklein, H. (1996): Groups as Galois groups. Cambridge University Press, Cam-bridge

Volklein, H. (1998): Rigid generators of classical groups. To appear in Math. Ann. Wagner, A. (1978): Collineation groups generated by homologies of order greater

than 2. Geom. Dedicata 7, 387-398 Wagner, A. (1980): Determination of the finite primitive reflection groups over an

arbitrary field of characteristic not 2, I-III. Geom. Dedicata 9, 239-253; 10, 183-189; 10, 475-523

Walter, J.H. (1984): Classical groups as Galois groups. Pp. 357-383 in: M. Asch-bacher et al. (eds.) (1984)

Ward, H.N. (1966): On Ree's series of simple groups. 'fians. Amer. Math. Soc. 121, 62-89

Washington, L. (1982): Introduction to cyclotomic fields. Springer, New York Weigel, T. (1992): Generators of exceptional groups of Lie type. Geom. Dedicata

41,63-87 Weil, A. (1956): The field of definition of a variety. Amer. J. Math. 78, 509-524 Weil, A. (1974): Basic number theory. Springer, New York Weissauer, R. (1982): Der Hilbertsche Irreduzibilitatssatz. J. reine angew. Math.

334, 203-220 Wilson, R.A. (1987): Some subgroups of the Baby Monster. Invent. Math. 89, 197-

218 Wolf, P. (1956): Algebraische Theorie der galoisschen Algebren. VEB Deutscher

Verlag der Wissenschaften, Berlin Yakovlev, A.V. (1964): The embedding problems for fields. Izv. Akad. Nauk. SSSR

28, 645-660 (Russian) Yakovlev, A.V. (1967): The embedding problem for number fields. Izv. Akad. Nauk.

SSSR 31,211-224 (Russian) [English trans!': Math. USSR Izv. 1, 195-208 (1967)J Zalesskii, A.E., Serezkin, V.N. (1980): Finite linear groups generated by reflections.

Izv. Akad. Nauk SSSR Ser. Math. 44, 1279-1307 (Russian) [English trans!.: Math. USSR Izv. 17 (1981), 477-503J

Zassenhaus, H. (1958): Gruppentheorie. Vandenhoeck & Ruprecht, Gottingen

Index

accompanying Brauer embedding problem 329

accompanying embedding problem 329

admissible covering 364 admissible subset 364 affinoid analytic space 364 algebraic fundamental group 4, 187 almost character 124 arithmetic fundamental group 9,196 (full) Artin braid group 178 AV-rigid 63 AV -symmetric 63 A V -symmetrized irrationality degree

63 axis of a pseudo-reflection 251

Belyi triple 99 braid relations 180 Brauer embedding problem

center of a pseudo-reflection central embedding problem clean Belyi function 17 closed ultrametric disc 374 coherent sheaf 367

317

251 266

cohomologically trivial in dimension i 339

compatible family 379 concordance obstruction 342 concordant embedding problem 329 conformal orthogonal group 109 conformal symplectic group 106 connected rigid analytic space 365 coroot 91 cyclotomic character cyclotomic polynomial

14 115

disclosed function field of one variable 13

embedding problem 265 existentially closed 386

field of definition 18 field of definition with group 18 finite embedding problem 266 finite morphism 372 first embedding obstruction 342 Frattini embedding problem 266 full symmetry group 31,62

G-compatible family 381 G-realization 33 GA-realization 35 GAR-realization 278 general unitary group 104 generating s-system 26 geometric (proper) solution 266 geometric embedding problem 266 geometric field extension 8 geometrically conjugate 123 gluing datum 364 good reduction modulo p 87 Green function 125 group of geometric automorphisms

Hi -rigid class vector 210 Hasse embedding obstruction 342 Hasse-Witt-invariant 309 Hilbertian field 264 Hilbertian set 264 homology 97 homomorphism ramified in 392 (full) Hurwhz braid group 180

induced cover 374 irrationality degree 28

j-th braid orbit genus 211

k-rational class vector 296

41

436 Index

k-symmetric class vector 296 kernel of an embedding problem 266

large field 388 Lusztig series 123

M-section 381 morphism of rigid analytic spaces 364

non-split embedding problem 266 normalized structure constant 36

open ultrametric disc 374 orthogonal group 107 orthogonal group of minus type 113 orthogonal group of plus type 109

p-stable 85 pairwise adjusted 383 parametric solution of an embedding

problem 266 primitive linear group 97 primitive prime divisor 115 primitive translate 53 profinite Hurwitz braid group 188 projective profinite group 272 proper solution (field) of an embedding

problem 266 pseudo algebraically closed 225 pseudo-reflection 97 pure Artin braid group 178 pure Hurwitz braid group 180

quasi-central element 129 quasi-determinant 112 quasi-p-group 396

r-fold uncomplete product 178 r-fold uncomplete symmetric product

178 rational class vector 28 rational subset 363 rationally rigid class vector 28 reduced braid orbit genera 241 reflection 97 rigid analytic space 364 rigid braid cycle 243 rigid class vector 28 rigid H~ -orbit 47,63 rigid H. -orbit 210 ring of holomorphic functions 363

root 90

s-th V -symmetrized braid orbit genus 231

Scholz embedding problem 352 Scholz extension 352 Scholz solution 352 Schur multiplier 222 second embedding obstruction 342 semiabelian group 276 semirational class 41 shape function 222 socle of an i-Galois extension 352 solution field of an embedding problem

266 solution of an embedding problem 265 sphere relations 181 spinor norm 107 split embedding problem 266 strictly non-degenerate quadratic form

310 symmetry group symplectic group

t.i.-property 155 Tate algebra 362

31,62 106

thick normal subgroup 184 transference 326 transvection 97 trivial cover 374 twisted structure sheaf 368

uniform function 123 unipotent character 123 uniquely liftable 295 unirational function field 199 universally central embeddable Galois

extension 306 unramified 186 unramified rational place 219

V-configuration 47,228 V-rigid class vector 47 V -symmetric 31 V -symmetrized braid orbit 209 V-symmetrized irrationality degree

31 Volklein's base group 249

wreath extension 326

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