Appendix: Example Polynomials

In this appendix we list example polynomials whose roots generate regular exten- sion 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 equiva- lence 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 31 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 14. 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 computer 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 of degree 12 to 14 are taken from Kl¨uners and Malle (2000, 2002). The polynomials listed in this table were chosen so that their coefficient sum is small.

1 Regular Realizations for Transitive Groups of Degree Less than 12

Here we give polynomials generating regular field extensions of Q.t/ with Galois 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 numbered 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 loc. cit.

© Springer-Verlag GmbH Germany, part of Springer Nature 2018 491 G. Malle, B. H. Matzat, Inverse Galois Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-662-55420-3 492 Appendix: Example Polynomials

Table 1.1 Symmetric and alternating groups n Sn (x  t.nx n C 1/ n.n1/=2 2 fSn .x;1  .1/ nt / for n Á 1.mod 2/ An 1 n.n1/= 2 fSn .x; =.1 C .1/ 2.n  1/t // for n Á 0.mod 2/

Table 1.2 Degree 4 4 2 f4;3 x  2x C t 4 2 f4;2 x C tx C 1 4 3 2 f4;1 x C tx  6x  txC 1

Table 1.3 Degree 5 5 3 2 2 f5;3 x C 10x C 5tx  15x C t  t C 16 2 2 2 f5;2 x .x C 1/ .x C 2/  .x  2/ .x  1/t 2 2 4 3 2 2 2 4 f5;1 x.x  25/ C .x  20x  10x C 300x  95/t  4.x  3/ t

Table 1.4 Degree 6 6 5 2 f6;14 x  2x C .5x  6x C 2/t 6 2 f6;13 x  .3x  2/ t 2 f6;12 f6;14.x;1  5t / 6 2 f6;11 x  .3x C 1/t=4 2 f6;10 f6;13.x;1=.t C 1// 2 f6;9 f6;13.x;1  t / 2 f6;8 f6;11.x;1=.1  3t // 2 f6;7 f6;11.x;t / 2 f6;6 f6;11.x;3t C 1/ 2 2 f6;5 f6;13.x;12t .3t C 1// 2 2 2 2 f6;4 f6;11.x;.t C 3/ =.t  3/ / 2 2 2 f6;3 x .x C 3/ C 4t 2 f6;2 f6;3.x;3t C 1/ 6 2 2 2 f6;1 x C .3x C 4/ .3t C 1/

Table 1.5 Degree 7 4 3 3 2 f7;5 .x  3x  x C 4/.x  x C 1/  x .x  1/t 7 6 5 3 2 6 4 f7;4 x C 28x C 63x C 1890x C 3402x  5103x C 33534 C x.x  63x 3402x  5103/t C 13122t 2

f7;3 see Smith (1993) 3 2 3 2 f7;2 f7;4 .x;.t  27t  9t C 27/=.3.t C t  9t  1/// 1 Regular Realizations for Degree Less than 12 493

7 5 3 2 4 6 5 4 3 f7;1 x 217.t/x 77.t/.10t C 5t 5t 3/x 7.15t C 15t 20t 27t 2 3 9 8 7 6 5 4 3 13t 6t 13/7.t/x 7.12t C 18t 30t 63t 35t 14t 35t 2 2 11 10 9 8 7 C2t C 31t C 16/7.t/x 7.t 1/7.t/.5t C 15t 5t 62t 93t 91t 6 126t 5 166t 4 113t 3 30t 2 8t 12/x .6t 15 C 15t 14 35t 13 126t 12 63t 11 C 70t 10 91t 9 271t 8 C 131t 7 C 427t 6 C 126t 5 84t 4 3 2 C175t C 189t 29t 97/7.t/

Table 1.6 Degree 8 4 2 2 2 2 f8;48 x .x  2/ .x C x C 2/  .x  1/ .x C x C 1/t 8 2 f8;47 x  .4x  3/ t 2 f8;46 f8;47.x;t C 1/ 2 f8;45 f8;47.x;1=.1  t // 8 2 f8;44 x C .4x C 3/t 6 2 f8;43 x .x  x C 7/  108.x  1/t 2 2 f8;42 f8;47.x;1=.12t .3t  1/// 2 4 6 2 3 f8;41 .x  2/  2 .2x  3/ t=3 4 4 2 f8;40 x .x  8x C 18/  27t 2 f8;39 f8;44.x;3t / 2 f8;38 f8;44.x;1=.3t C 1// 8 7 2 2 f8;37 x C 6x C 3.7x C 6x C 36/.7t C 144/

f8;36 .f9;32.x;0/  f9;32.t;0//=.x  t/ 2 f8;35 f8;44.x;27t .t  1/=4/ 2 f8;34 f8;41.x;1  t / 2 f8;33 f8;41.x;1=.3t C 1// 2 f8;32 f8;40.x;3t / 2 2 2 3 f8;31 f8;44.x;27.t  1/ =.t C 3/ / 4 4 2 2 2 2 2 f8;30 x .x C 4x C 6/  .4x C 1/.3t C 2/ .3t  1/=4 2 2 2 f8;29 f8;44.x;27t .t  1/ =4/ 4 2 f8;28 f8;44.x;27t .t C 1/=4/ 2 2 3 f8;27 f8;44.x;27.t C 4/=.4.t C 3/ // 2 2 f8;26 f8;40.x;27t .t  1/=..3t C 1/.3t  2/ //

f8;25 see Smith (1993) 2 4 2 f8;24 .x C x C 1/  .2x C 1/ t 2 2 2 2 2 2 2 f8;23 x .x C 396/ .x C 11/  .x C 4/ .x C 256/t 2 2 3 f8;22 f8;41.x;27t =.4.t  1/ // 8 2 6 4 2 4 6 4 2 8 6 f8;21 x C 2.t  1/x C .3t  t /x C 2.t C t /x C t C t 4 2 f8;20 f8;41.x;27t .t C 1/=4/ 494 Appendix: Example Polynomials

2 2 3 f8;19 f8;41.x;27.t C 4/=.4.t C 3/ // 2 2 2 3 f8;18 f8;41.x;27.t  1/ =.t C 3/ / 4 3 2 2 2 2 2 f8;17 .x C 4x  6x  4x C 1/  16x .x  1/ t 2 2 4 2 f8;16 f8;40.x;27.t C 2/ =.t .4t C 9/// 8 6 4 2 2 f8;15 x C 8x C 4.4t  11/x C 8.t  3/.t  2/x C t.t  3/ 2 f8;14 f8;24.x;1  3t / 2 f8;13 f8;24.x;1=.3t C 1// 2 f8;12 f8;23.x;t / 2 f8;11 f8;15.x;t / 2 3 4 2 2 2 6 f8;10 f8;41.x;2 3 t .t C 9/.t C 1/=.t C 3/ / 2 f8;9 f8;24.x;27t .t  1/=4/ 2 2 f8;8 f8;15.x;.8t C 3/=.2t C 1// 2 f8;7 f8;15.x;4=.t C 1// 2 f8;6 f8;15.x;2t C 3/ 8 2 2 6 2 2 2 2 4 f8;5 x  4.t C 2/.t C 1/x C 2.3t C 1/.t C 1/.t C 2/ x 4.t 2 C 2/2.t 2 C 1/3x2t 2 C .t 2 C 2/2.t 2 C 1/4t 4 4 2 2 2 2 2 f8;4 .x  6x C 1/ C 16x .x  1/ t 2 2 2 3 f8;3 f8;24.x;27.t  1/ =.t C 3/ / 2 2 3 f8;2 f8;24.x;27.t C 4/=.4.t C 3/ // 8 4 6 2 4 4 2 4 2 2 4 4 f8;1 x  4.t C 1/x C 2.4t C 1/.t C 1/x  4.t C 1/.t C 1/t x C .t C 1/t

Table 1.7 Degree 9 9 8 7 6 5 4 3 2 f9;32 x  3x C 4x  28x C 126x  266x C 308x  tx C .3t  539/x 4t C 805 4 3 2 3 f9;31 x .x C 1/ .x C 3/  4=27.3x C 1/ t 4 2 3 2 8 f9;30 x .x  3/ .x  3x  12/ C 2 t 2 f9;29 f9;31.x;t / 2 f9;28 f9;31.x;1=.3t C 1// 3 2 3 f9;27 f9;32.x;.t  6t C 3t C 1/=.t  3t C 1// 3 2 2 3 2 f9;26 .x  19x C 97x  27/.x  4x  7/ C 16=27x .x  7/t 2 f9;25 f9;30.x;1=.3t C 1// 6 3 3 f9;24 x .x C 9x C 6/  4.3x C 2/ t 2 2 f9;23 f9;26.x;.43923t C 18225/=.3t C 1// 2 f9;22 f9;24.x;1=.3t C 1// 4 2 2 5 2 2 9 2 f9;21 x .x C 1/ .x C 2/ .x C 3/  1=3 x .9x C 20x C 12/t C 1=3 t 2 f9;20 f9;24.x;3t C 1/ 2 2 f9;19 .f10;35.x;0/.t  1/  f10;35.t;0/.x  1/ /=.x  t/ 6 2 3 f9;18 x .x C 1/ .x  2/ C 4=27.3x C 2/ t 5 2 f9;17 f9;21.x;2 =.3t C 1// 1 Regular Realizations for Degree Less than 12 495

2 4 4 3 3 f9;16 .x C x  2/ .x  4/ C 2 3 x t

f9;15

f9;14 2 f9;13 f9;18.x;1=.3t C 1// 2 f9;12 f9;18.x;t / 2 f9;11 f9;18.x;3t C 1/ 6 4 3 3 2 f9;10 .x C 3x C 10x C 6x C 25/.x C 3x C 2/ C .x  1/.x C 2/ .x3  3x2  6x  1/  .x4  7x3 C 6x2  13x  14/t=9 C 18t 2 2 f9;9 f9;16.x;t C 1/ 2 f9;8 f9;16.x;t / 3 2 2 3 2 2 2 2 f9;7 .x C 27x  9x  27/.x C 3/  27=4.x  1/ x.x  9/.3t C 49/ 2 2 2 4 2 2 f9;6 f9;21.x;96.t  9/ t =..t  2t C 49/.3t C 1/// 2 2 2 2 f9;5 f9;16.x;.t C 1/ =.t  1/ / 2 2 f9;4 f9;16.x;1=.3t C 1/ / 3 2 3 f9;3 f9;10.x;.t C 6t C 3t  1/=.t  3t  1// 9 2 7 6 4 2 5 2 4 f9;2 x  6.t C 3/x  6x t C 9.t C 9t C 9/x C 24t.t C 3/x .4t 6 C 69t 4 C 213t 2 C 81/x3  216t 3x2 C 12t 2.3t 4  11t 2 C 21/x  8t 3 9 7 2 6 4 3 2 5 f9;1 x  279.t/x  54t.t  1/9.t/x C 2439.t/.2t C t  t C 1/x 2 4 3 2 4 8 7 6 5 C243t.t 1/9.t/  .4t C2t t CtC3/x 81.33t C33t 26t 6t 4 3 2 3 2 8 7 6 5 C69t C16t 36t 3tC10/  9.t/x 2187t.t 1/.2t C2t t t 4 3 2 2 3 9 8 7 6 4 C4t C3t t C1/9.t/x C729.2t C1/.3t C9t C2t 14t C17t 3 2 13 12 11 10 9 Ct 9t tC1/9.t/xC2439.t/.36t C18t 60t C30t C64t 81t 8 9t 7C87t 6 36t 5 54t 4C21t 3C15t 2 3t 1/

Table 1.8 Degree 10 10 2 f10;43 x  .5x  4/ t 2 f10;42 f10;43.x;1=.t C 1// 2 f10;41 f10;43.x;1  t / 2 2 f10;40 f10;43.x;20t .5t  1// 10 5 2 f10;39 x  5 .x C 4/t 2 f10;38 f10;39.x;1=.t C 1// 2 5 5 2 f10;37 .x  4/  5 x t 2 f10;36 f10;39.x;1  5t / 10 9 8 2 f10;35 x  2x C 9x  729.x  1/ t 2 2 2 2 f10;34 f10;39.x;4.t C t  1/ =.5.t C 1/ // 2 2 4 6 5 4 3 2 f10;33 .x  2/ .x C x  1/  .380x  784x C 300x C 360x  315x C60x C 4/t C 4.5x  4/2 t 2 2 f10;32 f10;35.x;t / 2 f10;31 f10;35.x;1=.2t C 1// 496 Appendix: Example Polynomials

2 f10;30 f10;35.x;1  2t / 10 6 4 2 2 f10;29 x C 10x  5tx  15x  t C t  16 4 3 2 2 2 5 2 f10;28 .x  2/.x  1/.x C x C 6x  4x C 1/.x C x  1/ C .4x  20x C15x  2/  .10x3  10x2 C 1/t C .5x  4/.8x5  40x2 C 35x  8/t 2 2 f10;27 f10;33.x;t=.4.t  1/// 2 2 2 2 f10;26 f10;35.x;.t  2/ =.t C 2/ / 8 2 4 2 2 4 f10;25 f10;39.x;2 t =..t C 6t C 25/.t C 1/ // 2 f10;24 f10;29.x;t.t  8/=.t  1// 2 4 2 8 6 4 2 2 2 2 f10;23 x .x  25/ C .x  20x  10x C 300x  95/t  4.x  3/ t 10 5 2 4 f10;22 x C 5 .x C 256/ t 2 2 2 2 2 5 4 2 f10;21 .x C 9x C 24/ .x C 4x C 64/ .x  6x C 144/  5 x .x C 8/ t=4

f10;20 2 4 2 7 5 4 3 2 f10;19 .x C 1/ .x C 16/  5.x C 11x  15x  5x C 38x  15x  7/t C.x5 C 10x3  15x  15x2 C 28/t 2 2 f10;18 f10;28.x;t /

f10;17 2 f10;16 f10;23.x;t  95=36/ 2 f10;15 f10;23.x;95=.t  36// 2 2 f10;14 f10;23.x;t / D f5;1.x ;t/ 2 5 5 2 4 f10;13 .x  5/  5 .x C 5x C 6/ t=4 2 f10;12 f10;22.x;1=.1  t // 2 f10;11 f10;22.x;1  5t / 2 2 f10;10 f10;21.x;4.3t C 32/=.4t C 1// 2 f10;9 f10;21.x;4.5t  32// 2 2 2 2 f10;8 f10;23.x;.7t  24t C 7/ =.36.t  1/ // 2 f10;7 f10;13.x;1  5t / 4 2 f10;6 f10;19.x;4=.5t C 5t C 1// 5 5 2 f10;5 f10;22.x;4t .t  10/=.5 .t C 2t C 5/// 5 5 2 f10;4 f10;13.x;4t .t  10/=.5 .t C 2t C 5/// 2 2 5 5 2 2 2 4 f10;3 f10;22.x;.11t C 4t  11/.t C 4t  1/ =.5 .t C 1/ .t  1/ // 10 2 8 2 2 6 2 2 4 f10;2 x  2.t  125/x C .t  125/.t  4t  65/x  4.t  125/ .t  10/x C4.t 2  14t C 25/.t 2  125/2x2  64.2t  25/.t 2  125/2 10 8 4 3 2 6 f10;1 x  2010.t/x C 10.7t  7t C 17t  17t C 12/10.t/x 8 7 6 5 4 3 2 4 25.4t  8t C12t 16t C25t 46t C67t 38tC9/10.t/x 12 11 10 9 8 7 C510.t/.13t 39t C18t C50t 125t C376t 453t 6 214t 5C1050t 4 1125t 3C613t 2 164tC18/x2 2 3 4 5 6 7 8 2 10.t/.13tC32t 36t 10t C34t 13t 8t C4t / 1 Regular Realizations for Degree Less than 12 497

Table 1.9 Degree 11 f .f .x;0/.2t  1/2  f .t;0/.2x  1/2/=.x  t/ 11;6 M12 M12 11 10 9 8 7 6 4 3 2 f11;5 x  3x C 7x  25x C 46x  36x C 60x  121x C 140x  95x C 27 Cx2.x  1/3 t

f11;4 11 2 9 2 2 7 2 3 5 2 4 3 f11;3 x  11.t C 11/x C 44.t C 11/ x  77.t C 11/ x C 55.t C 11/ x 11.t 2 C 11/5x  2t.t2 C 11/5

f11;2 11 9 5 4 3 2 8 f11;1 x  5511 .t/x  11.30t C 15t  30t  25t  4t C 3/11.t/x 11.90t 10C90t 9240t 8350t 7229t 697t 5C35t 4C13t 342t 2 7 15 14 13 12 11 42t75/11.t/x 11.168t C252t 840t 1750t 1218t 242t 10C880t 9C1265t 8C880t 7C836t 6C572t 5C437t 4C430t 3 2 6 20 19 18 17 C224t C78t36/11.t/x 11.210t C420t 1680t 4550t 2723t 16C2118t 15C7971t 14 C11976t 13C9282t 12 C6555t 11C6523t 10 C5466t 9C6103t 8C4089t 7422t 62128t 51887t 4722t 3C355t 2 5 25 24 23 22 C508tC452/11.t/x 11.180t C450t 2100t 7000t 3080t 21C10615t 20 C27060t 19 C40865t 18C32857t 17 C10109t 16 2398t 15 10128t 14 6994t 13 882t 1214413t 11 33099t 10 42438t 9 36861t 818117t 7550t 6C6589t 5C2640t 41063t 3958t 2648t 4 30 29 28 27 26 C117/11.t/x 11.105t C315t 1680t 6650t 1659t C20003t 25 C44905t 24 C64445t 23C44116t 22 34353t 21 102124t 20 135499t 19 138713t 18 92626t 17 79067t 16 119189t 15 147399t 14 166843t 13 136359t 12 38237t 11 C44396t 10 C74899t 9 C52267t 8 C2031t 722096t 612051t 5 C3115t 4C7001t 3C1543t 21896t 3 35 34 33 32 31 1160/11 .t/x 11.40t C140t 840t 3850t 154t C19008t 30 C39600t 29 C49203t 28 C14520t 27120945t 26 280357t 25 348952t 24 314514t 23 145540t 22 C29359t 21 C33825t 2027126t 19 75933t 18 85096t 17 C57717t 16 C276738t 15 C420112t 14 C438965t 13 C296100t 12 C50632t 11 97383t 1069608t 9C16104t 8 C68277t 7 6 5 4 3 2 2 C54527t 3025t 25355t 7986t C3117t C2302t84/11.t/x 11.9t 40 C36t 39240t 381250t 37C227t 36 C9128t 35 C17905t 34 C16150t 33 12716t 32122980t 31 290048t 30 377822t 29 311551t 28 C1083t 27 C489620t 26 C744371t 25 C662921t 24 C433805t 23 C172463t 22 C209836t 21 C561407t 20 C810964t 19 C909892t 18 C777874t 17 C289801t 16 197823t 15 430310t 14 356065t 13 11405t 12 C252280t 11 C230131t 10 C47388t 9 93665t 890187t 7 24467t 6 5 4 3 2 C19479t C19576t 4165t 5861t C1587tC999/  11.t/x .10t 45C45t 44330t 431925t 42 C792t 41C19448t 40 C36036t 39 C13761t 38 83787t 37 449020t 36 1138951t 35 1569333t 34 1270152t 33 C131912t 32 C3449677t 31 C7101292t 30 C8022157t 29 498 Appendix: Example Polynomials

C6359584t 28 C 2562879t 27  1238875t 26 C 266530t 25 C 4792381t 24 C7758954t 23 C 9292575t 22 C 6341588t 21 C 107481t 20  3610200t 19 4999456t 18  3552868t 17 C 1494614t 16 C 4899972t 15 C 3760834t 14 C620191t 13  2831935t 12  4464948t 11  2695792t 10  670956t 9 36608t 8 C 325281t 7 C 187935t 6 C 13585t 5 C 170786t 4 C 81906t 3 2 42372t  19548t C 243/11.t/ 2 Regular Realizations for Nonsolvable Primitive Groups 499 2 Regular Realizations for Nonsolvable Primitive Groups

Here we collect regular realizations for primitive non-solvable permutation groups of degree d with 12 Ä d Ä 31. Simple groups in this range for which no polynomial over Q.t/ is known to date are L2.16/,M23,L2.25/ and L2.27/. The polynomials were taken from Hafner¨ (1992), Konig¨ (2015), Malle (1987, 1988a, 1993a), Malle and Matzat (1985), Matzat (1987), Matzat and Zeh-Marschke (1986)andM¨uller (2012). We also present surprisingly small polynomials of degree 100 with groups Aut.HS/ and HS taken from Barth and Wenz (2016). A polynomial of degree 266 for the Janko group J2 has been obtained in Barth and Wenz (2017). In addition we give the polynomial with Galois group Z16 from Dentzer (1995a).

Table 2.1 Primitive groups 12 11 10 9 8 7 6 M12 x C 44x C 754x C 6060x C 18870x  28356x  272184x 57864x5 C 1574445x4  92960x3  1214416x2 C 1216456x 304119  492075.2x  1/2 t 3 4 5 4 3 2 PGL2.11/ .x  66x  308/  9t.11x  44x  1573x C 1892x C 57358x C103763/  3t 2.x  11/ .11/ f .x;2835=.11t 2 C 1// L2 PGL2.11/ 6 4 3 2 4 3 2 L3.3/ .x  6x C 64x  36x C 216/.x C 8x  108x C 432x  540/ .x3  18x2 C 54x  108/  .3x4  28x3 C 108x2  216x C 108/2 .x4 C 8x3 C 108/t 3 2 4 2 2 6 2 PGL2.13/ .x  x C 35x  27/ .x C 36/  4.x C 39/ .7x  2x C 247/t=27 .13/ f .x;1=.39t 2 C 1// L2 PGL2.13/ 3 2 6 17 15 14 13 12 11 PGL2.17/ .x  7x C 5x  2/  .x  17x C 34x C 85x  408x C 289x C1190x10  2907x9 C 1462x8 C 3281x7  5780x6 C 3196x5 C 238x4 646x3  68x2 C 120x  16/t C t 2 .17/ f .x;223317=.t 2  17// L2 PGL2.17/ 5 4 3 2 4 19 17 16 PGL2.19/ .x C 26x C 69x C 108x C 68x C 16/  .x  38x  38x C513x15 C 1064x14  2299x13  9538x12  5358x11 C 24358x10 C55081x9 C 35416x8  40204x7  105374x6  98496x5  41040x4 C3648x3 C 11552x2 C 4352x C 512/t C t 2 .19/ f .x;2819=.t 2 C 19// L2 PGL2.19/ 3 2 5 5 5 4 3 2 P€L3.4/ .x  9x  21x C 5/ .x C 1/ x  t.20x C 89x C 68x  50x C16x C 1/3  .x5 C 57x4 C 330x3 C 914x2 C 1509x C 1125/

L3.4/:3 see Konig¨ (2015) .4/:2 .f .x;0/.t 2  t C 3/11  .f .t;0/.x2  x C 3/11/=.t  x/ L3 2 Aut.M22/ Aut.M22/ L3.4/ see Konig¨ (2015) 500 Appendix: Example Polynomials

4 3 2 4 3 2 2 Aut.M22/ .5x C 34x  119x C 212x  164/ .19x  12x C 28x C 32/ 222.x2  x C 3/11 t f .x;1=.11t 2 C 1// M22 Aut.M22/ 10 9 8 7 6 5 4 3 M24 4.48x  192x  256x C 1104x C 520x  1276x  64x  776x 1117x2 C 391x C 52/2.x2 C 1/ C .16x12  96x11  144x10 C 928x9 C520x8  1744x7  1008x6  1712x5  791x4 C 2154x3 C 1121x2 C1098x  t/2 8 7 6 5 4 3 2 3 PGL2.23/ .x C 3x C 37x  24x C 121x C 333x C 429x C 216x C 36/ .2x24Cx23  322x22C1219x21C1863x20C4094x19C99084x18 C197501x17C877910x16C1337726x15 C3132117x14 C8697795x13 C15394935x12 C16590866x11 C4182642x10 C6982731x9 C36934642x8C43085601x7 C13510591x6  9423054x5 10152936x4  4024080x3  824688x2  85536x  3456/t C.x24  7x23C69x22  460x21  1564x20  3289x19C11017x18 C19159x17  20792x16  269307x15  650440x14  547124x13 C609937x12C2106294x11 C2682306x10 C1410682x9  856612x8 1557215x7  609132x6C135079x5C225814x4C113436x3 C33764x2C5904xC496/t 2  .x23C23x20C23x19C23x18 C161x17C368x16C529x15C575x14C1610x13C3036x12 C2668x11C2300x10C3542x9C5428x8C2599x7  1748x6 1265x5C345x4  598x2  252x  16/t 3Ct 4 .23/ f .x;.23  33t 2/=.t 2 C 23// L2 PGL2.23/ 3 2 9 4 12 6 2 4 U4.2/:2 .x C 6x  8/  2 3 x .x C 5x C 4/ .x  2/t 3 2 9 4 12 6 2 4 2 U4.2/ .x C 6x  8/  2 3 x .x C 5x C 4/ .x  2/.3t C 1/ 4 2 7 3 2 5 S6.2/ .x  10x  8x C 1/  x .x C 3x C 1/ t 6 5 4 3 2 4 4 3 2 U3.3/:2 .x  6x  435x  308x C 15x C 66x C 19/ .x C 20x C 114x C68x C 13/  2239.x2 C 4x C 1/12.2x C 1/t .3/ f .x;1=.t 2 C 1// U3 U3.3/:2 5 4 3 2 6 29 26 25 24 PGL2.29/ .x  7x C 8x  17x C 9x  6/  t.x C 29x  29x C 29x C290x23  638x22 C 899x21 C 464x20  4118x19 C 8323x18 9686x17  899x16 C 20532x15  46197x14 C 55477x13  36801x12 8584x11 C 66874x10  100601x9 C 105560x873602x7 C 34017x6 2349x5  11745x4 C 10962x3  6264x2 C 1944x  432/ C t 2 .29/ f .x;223329=.t 2  29// L2 PGL2.29/ 5 4 3 2 3 5 4 3 2 PSL5.2/ .x  95x  110x  150x  75x  3/ .x C 4x  38x C 56x C53x  4/3.x  3/  34t.x2  6x  1/8.x2  x  1/4.x C 2/4x Aut.HS/ .x4  5/5.x8  20x6 C 60x5  70x4 C 100x2  100x C 25/10 t.7x5  30x4C30x3C40x2  95xC50/4.2x10  20x9C90x8  240x7 C435x6  550x5C425x4  100x3  175x2C250x  125/4.2x10C5x8 40x6C50x4  50x2C125/4 2 8 HS fAut.HS/.x;.5t C 1/=2 / 2 Regular Realizations for Nonsolvable Primitive Groups 501

Table 2.2 The cyclic group Z16 16 4 14 4 6 4 2 12 Z16 x  2 16.t/x C 2 .16t  14t C 6t C 5/16.t/x 6 12 10 8 6 4 2 10 2 .24t  28t C6t C36t  31t C13t C2/16.t/x C25.128t 18  120t 16  144t 14C560t 12  488t 10C144t 8C164t 6 4 2 8 136t C56t C1/16.t/x 28.16t 22C16t 20  120t 18C208t 16  108t 14  64t 12C164t 10 8 6 4 2 2 6 128t C73t  20t C3t C2/t 16.t/x C28.64t 24  192t 22C208t 20C80t 18  432t 16 14 12 10 8 6 4 2 4 4 C520t  316t C112t C18t  66t C67t  26t C5/t 16.t/x 210.32t 22  112t 20C160t 18  72t 16  84t 14C144t 12  86t 10C28t 8 6 4 2 6 2 17t C17t  7t C1/t 16.t/x 8 10 8 6 2 2 8 C2 .8t  16t C12t  4t C1/ t 16.t/ 502 Appendix: Example Polynomials 3 Realizations over Q for Transitive Groups of Degree up to 14

This last set of tables contains polynomials generating field extensions of Q with transitive Galois group of degree less than fifteen. The polynomials are mainly taken from the database Kl¨uners and Malle (2002), which contains polynomials for all but two transitive groups up to degree 23; see also Kl¨uners and Malle (2000).

Table 3.1 Degree 2 2 T1 2 x C x C 1

Table 3.2 Degree 3 3 T2 S3 x  x  1 3 2 T1 3 x  x  2x C 1

Table 3.3 Degree 4 4 T5 S4 x  x C 1 4 3 2 T4 A4 x  2x C 2x C 2 4 3 2 T3 D4 x  x  x C x C 1 4 2 T2 V4 x  x C 1 4 3 2 T1 4 x C x C x C x C 1

Table 3.4 Degree 5 5 3 2 T5 S5 x  x  x C x C 1 5 4 2 T4 A5 x C x  2x  2x  2 5 4 3 2 T3 F20 x C x C 2x C 4x C x C 1 5 3 2 T2 D5 x  x  2x  2x  1 5 4 3 2 T1 5 x C x  4x  3x C 3x C 1

Table 3.5 Degree 6 6 4 3 T16 S6 x  x  x C x C 1 6 3 2 T15 A6 x  x  3x  1 6 5 T14 PGL2.5/ x  2x C 4x C 2 2 6 5 2 T13 3 :D4 x C x  x  x C 1 6 5 2 T12 L2.5/ x  2x  5x  2x  1 6 4 T11 2  S4 x  x C 1 2 6 5 4 3 2 T10 3 :4 x C x C x C x  4x C 5 2 2 6 3 T9 3 :2 x  x C 2 6 4 2 T8 S4=4 x  x C 2x C 2 6 2 T7 S4=V4 x  x  1 6 2 T6 2  A4 x  3x C 1 6 3 T5 3  S3 x  3x C 3 6 4 2 T4 A4 x C x  2x  1 6 3 T3 D6 x  x  1 6 T2 S3 x C 3 6 3 T1 6 x  x C 1 3 Realizations over Q for Transitive Groups 503

Table 3.6 Degree 7 7 3 2 T7 S7 x C x  x C 1 7 6 T6 A7 x  2x C 2x C 2 7 T5 L3.2/ x  7x C 3 7 T4 F42 x  2 7 5 4 3 2 T3 F21 x  8x  2x C 16x C 6x  6x  2 7 3 2 T2 D7 x C 7x  7x C 7x C 1 7 6 5 4 3 2 T1 7 x  x  12x C 7x C 28x  14x  9x  1

Table 3.7 Degree 8 8 4 T50 S8 x C x C x C 1 8 3 T49 A8 x  8x C 10 3 8 7 T48 2 :L3.2/ x  2x C 8x  2 8 T47 S4 o 2 x  5x  5 8 3 2 T46 x  8x  8x C 1 8 4 2 T45 x  3x  2x  4x  1 8 2 T44 2 o S4 x  x  1 8 7 6 T43 PGL2.7/ x  x C 7x  4x C 4 8 7 4 T42 A4 o 2 x  2x C 6x C 4 8 7 4 2 T41 x C 4x  2x  4x C 2 8 6 T40 x C 4x  9 3 8 2 T39 2 :S4 x C x C 1 8 6 4 T38 2 o A4 x C 2x C 2x C 2 8 7 6 5 4 3 2 T37 L2.7/ x  4x C 7x  7x C 7x  7x C 7x C 5x C 1 3 8 7 6 5 4 3 T36 2 :7:3 x C x C x  3x C 5x C 5x  7x C 9 8 6 T35 2 o 2 o 2 x C 2x C 2 8 7 6 5 4 3 T34 x  x C 2x  x  2x C 4x  6x C 4 8 5 4 2 T33 x  4x C 12x  8x C 12x C 9 8 6 2 T32 x C x C 3x C 4 2 8 6 2 T31 2 o 2 x C 4x  8x  1 8 6 4 T30 x  4x C 4x  2 3 8 6 2 T29 2 :D4 x  x C x C 1 8 6 T28 x C 4x C 2 8 4 2 T27 2 o 4 x  8x C 8x  2 8 4 T26 x C x C 2 3 8 7 6 5 4 3 2 T25 2 :7 x  4x C 8x  6x C 2x C 6x  3x C x C 3 8 2 T24 S4  2 x  4x C 4 8 4 2 T23 GL2.3/ x  6x  x  3 8 4 T22 x  x C 4 8 6 4 T21 x  2x C x C 5 8 6 4 2 T20 x  3x  x C 3x C 1 8 4 2 T19 x C 4x  4x C 1 2 8 6 2 T18 2 o 2 x  x C 2x C 1 8 4 T17 4 o 2 x  2x C 2 8 4 T16 x C 4x C 2 8 T15 x C 3 8 6 2 T14 S4 x C 4x C 4x C 4 8 6 4 2 T13 A4  2 x C 2x C 3x  3x C 1 504 Appendix: Example Polynomials

8 6 4 2 T12 SL2.3/ x C 9x C 23x C 14x C 1 8 T11 x C 9 8 6 4 2 T10 x  2x C 4x  3x C 1 8 4 T9 D4  2 x C 4x C 1 8 T8 x  2 8 4 2 T7 x  15x C 10x C 5 8 T6 D8 x C 2 8 6 4 2 T5 Q4 x C 12x C 36x C 36x C 9 8 4 T4 D4 x C 3x C 1 3 8 4 T3 2 x  x C 1 8 T2 4  2 x C 1 8 7 6 5 4 3 2 T1 8 x C x  7x  6x C 15x C 10x  10x  4x C 1

Table 3.8 Degree 9 9 5 2 T34 S9 x C x  x C 1 9 3 T33 A9 x  3x C x C 2 9 7 5 3 2 T32 €L2.8/ x C x C 2x C 4x  x C x C 1 9 8 2 T31 S3 o S3 x  x C 2x  x C 1 9 5 4 3 2 T30 x C 2x  4x C 4x  4x C x  1 9 6 5 2 T29 x  3x  5x C 5x  1 9 6 3 T28 S3 o 3 x  2x  4x C 3x C 1 9 7 6 4 3 2 T27 L2.8/ x C x  4x  12x  x  7x  x  1 2 9 7 6 5 4 3 2 T26 3 :GL2.3/ x  x C 5x C x  2x C 4x C 3x  x  1 9 6 5 4 3 T25 x  3x C 9x  9x  27x C 9x C 1 9 6 T24 x  2x  2 2 9 8 6 5 4 3 T23 3 :SL2.3/ x  3x C x C 15x  13x  3x C 4x  1 9 6 T22 x  3x C 3 9 3 T21 x  6x  6 9 6 3 T20 3 o S3 x  x  2x C 1 9 8 5 4 T19 x  3x C 18x C 18x  27x C 9 9 3 T18 x  x  1 9 8 7 6 5 4 2 T17 3 o 3 x C x  10x  14x C 20x C 36x  18x  8x  1 2 9 8 5 4 3 2 T16 3 :D4 x  x  x  x C 3x C 2x  1 2 9 8 7 5 4 3 2 T15 3 :8 x  4x C 8x  32x C 80x  104x C 80x  34x C 8 2 9 5 T14 3 :Q4 x  12x C 132x  128 9 3 T13 x  3x  1 9 8 5 3 2 T12 x  2x C x  3x C 4x  12x C 8 2 9 6 3 T11 3 :6 x  x C 5x C 1 9 T10 9:6 x  2 2 9 7 6 5 4 3 T9 3 :4 x C 2x  3x C x  x C 64x  x  1 2 9 3 T8 S3 x C 3x  1 2 9 8 7 5 4 3 2 T7 3 :3 x  3x  21x C 78x C 69x  21x  39x  12x  1 9 7 5 3 T6 9:3 x  14x C 63x  98x C 42x  7 2 9 6 3 T5 3 :2 x  3x  3x  1 9 6 3 T4 S3  3 x  3x  6x  1 9 6 3 T3 D9 x  9x C 27x  3 2 9 7 6 5 4 3 2 T2 3 x  15x C 4x C 54x  12x  38x C 9x C 6x  1 9 7 5 3 T1 9 x  9x C 27x  30x C 9x  1 3 Realizations over Q for Transitive Groups 505

Table 3.9 Degree 10 10 3 T45 S10 x  x  1 10 9 5 T44 A10 x  2x C 3x  4 10 6 5 T43 S5 o 2 x C 3x  2x C 1 10 8 7 6 5 4 2 T42 x C 5x  5x C 5x  7x  5x  10x  4 10 9 6 4 2 T41 x  2x  x C x  4x C 2x  1 10 9 4 3 2 T40 A5 o 2 x  x  x  4x C 4x  x  1 10 2 T39 2 o S5 x  x C 1 10 8 2 T38 x  3x C 2x C 2 4 10 2 T37 2 :S5 x  x  1 10 4 2 T36 2 o A5 x C x  2x C 3 10 9 8 2 T35 P€L2.9/ x  4x C 6x C 12x C 16x C 8 4 10 4 2 T34 2 :A5 x C 4x C x  4 10 6 5 2 T33 .5:4/ o 2 x C 6x C 8x  35x C 24x C 16 10 9 8 2 T32 S6 x  2x C x  9x C 2x  1 10 9 8 2 T31 M10 x  2x C 9x  54x C 108x  54 10 9 8 2 T30 PGL2.9/ x  2x C 9x  7x C 14x  7 10 6 T29 2 o .5:4/ x C 10x C 5 10 7 6 5 4 3 T28 x  10x C 10x C 36x C 50x  10x  1 10 6 5 2 T27 x C 3x  2x C x C 2x C 1 10 9 8 7 6 5 3 2 T26 L2.9/ x  x C 3x  6x C 3x  3x  3x  6x  8x  1 10 6 T25 x C 10x  5 4 10 6 2 T24 2 :5:4 x C 5x C 5x  1 10 4 T23 2 o .5:2/ x  5x  3 10 2 T22 S5  2 x C 4x C 4 10 6 5 4 2 T21 D5 o 2 x C x  2x  x C 3x  2x C 1 2 10 8 6 5 4 3 2 T20 5 :Q4 x  10x C 35x  4x  50x C 20x C 25x  20x  17 2 10 8 6 5 4 3 2 T19 5 :D4 x  10x C 35x  2x  50x C 10x C 25x  10x C 2 2 10 6 5 2 T18 5 :8 x C 60x  208x C 850x  8000x  4672 10 5 T17 x C x C 2 10 4 T16 x  5x C 15 4 10 4 2 T15 2 :5:2 x  5x  4x  1 10 8 6 4 2 T14 2 o 5 x C x  4x  3x C 3x C 1 10 9 8 6 5 4 3 2 T13 S5=D6 x  x  x C 3x  x  2x C 3x  x  x C 1 10 9 8 6 5 4 2 T12 S5=A4 x C 2x C 3x  x  2x  x C 3x C 2x C 1 10 8 2 T11 A5  2 x C x  4x C 4 2 10 5 T10 5 :4 x  2x  4 2 2 10 9 8 6 5 4 2 T9 5 :2 x  x  5x C 11x C 4x  10x C 25x C 5x  5 4 10 8 6 4 2 T8 2 :5 x  4x C 2x C 5x  2x  1 10 8 7 6 5 3 T7 A5 x  x  4x  3x  2x C 8x  2x  1 10 9 7 6 5 4 3 2 T6 5 o 2 x  x C 3x  3x C x C 5x  x C 2x C 3x C 1 10 T5 2  5:4 x C 2 10 T4 5:4 x  5 10 4 2 T3 D10 x  3x C 2x C 1 10 8 6 4 2 T2 D5 x C 5x C 15x C 20x C 25x C 15 10 9 8 7 6 5 4 3 2 T1 10 x C x C x C x C x C x C x C x C x C x C 1 506 Appendix: Example Polynomials

Table 3.10 Degree 11 11 6 4 T8 S11 x C x C x C 1 11 8 5 3 T7 A11 x  6x C 4x  3x C 2 11 10 7 6 5 4 T6 M11 x  4x C 60x  108x C 72x  360x C 3636x  1944 11 10 9 8 7 6 5 4 3 2 T5 L2.11/ x  2x C x  5x C 13x  9x C x  8x C 9x  3x  2x C 1 11 T4 F110 x  3 11 9 7 5 3 T3 F55 x  33x C 396x  2079x C 4455x  2673x  243 11 10 8 5 4 3 2 T2 D11 x  x C 5x C 8x C 6x  x C x C 3x C 1 11 10 9 8 7 6 5 4 3 T1 11 x C x  10x  9x C 36x C 28x  56x  35x C 35x C15x2  6x  1

Table 3.11 Degree 12 12 S12 x  x C 1 12 8 4 3 A12 x C 3x C 3x C 4x C 4 12 2 T299 x C x  2x C 1 12 2 T298 x  72x  120x  50 12 7 6 2 T297 x  2x C 7x C x  2x C 1 12 7 6 4 2 T296 x  x  7x  5x  x C x C 1 12 8 6 3 2 M12 x  375x  3750x  75000x C 228750x  750000x C 1265625 12 9 7 T294 x C 4x  6x C 2 12 6 2 T293 x  x  x  1 12 11 9 8 7 6 5 4 T292 x  3x C 5x  3x C 3x C 2x  6x  3x C 1 12 9 8 3 2 T291 x  12x  9x  64x  144x  108x  27 12 10 9 6 5 4 3 T290 x C 3x  x C 2x  3x C 9x  3x C 3x C 1 12 9 6 4 T289 x  4x C 2x C 4x C 1 12 11 10 4 3 2 T288 x  4x C 4x  50x C 120x  112x C 48x  8 12 8 6 T287 x  3x C 2x C 3 12 6 4 T286 x  x  3x  1 12 2 T285 x  4x C 4 12 9 8 3 2 T284 x  12x  9x C 64x C 144x C 108x C 27 12 9 6 5 3 2 T283 x  8x C 24x C 144x C 96x C 144x C 48 12 11 10 9 8 6 5 4 3 2 T282 x  x C 3x  x C 6x C 6x  2x C 7x C 4x C 4x C x C 1 12 11 10 8 7 5 4 3 2 T281 x  x C x  2x C 3x  3x C 3x  2x C 3x  x C 1 12 11 10 9 8 7 5 4 T280 x  4x C 6x  2x  5x C 6x  4x C 2x C 2 12 11 10 7 6 5 2 T279 x  4x C 4x C 4x  6x  4x C 36x C 36x C 9 12 8 6 4 3 2 T278 x C 20x  80x C 50x  320x  912x C 1280x C 800 12 6 2 T277 x C 3x C 3x C 4 12 6 5 4 3 2 T276 x C 192x  288x C 108x C 256x  576x C 432x  108 12 10 9 8 7 6 5 4 T275 x C x  9x C 11x  11x C 17x  7x C 2x C x C 1 12 8 6 3 T274 x  x C 2x  4x C 1 12 9 8 3 2 T273 x  12x C 9x C 192x  432x C 324x  81 12 11 10 9 8 7 6 5 4 3 M11 x C 6x C 15x C 28x C 36x C 6x  75x  108x C 18x C 82x C3x2  6x C 5 12 8 7 6 5 4 3 2 T271 x  135x  180x C 399x C 918x C 693x C 352x C 216x C 96x C 16 12 10 2 T270 x C x C 4x  1 12 10 8 6 4 3 T269 x  2x  6x C 14x C x  8x C 1 12 9 8 3 2 T268 x C 4x  3x  64x C 144x  108x C 27 3 Realizations over Q for Transitive Groups 507

12 10 9 8 7 6 5 4 3 T267 x C 12x  8x C 54x  48x C 132x  72x  33x  32x C 8 12 10 9 8 7 5 3 2 T266 x C x  4x  2x  3x C 4x C 2x  2x  x C 1 12 10 9 8 7 6 5 4 3 2 T265 x  8x C 7x C 8x  7x  15x C 21x C 8x  14x  8x C 7x C 1 12 11 10 9 8 7 5 4 3 2 T264 x  4x C 6x  3x  2x C 3x  2x C x C x  x C 1 12 4 3 2 T263 x  162x  432x  432x  192x  32 12 8 7 6 4 3 2 T262 x C 18x  24x C 8x  81x C 216x  216x C 96x  16 12 6 5 4 T261 x C 2x  4x C x C 1 12 2 T260 x  3x C 3 12 10 8 6 4 3 2 T259 x  12x C 54x  110x C 93x  4x  18x C 12x  8 12 3 T258 x  x  3 12 10 2 T257 x C 4x  4x C 4 12 10 2 T256 x C 4x  5x C 5 12 8 6 4 T255 x  2x  6x C 9x  1 12 10 9 7 6 4 3 2 T254 x C 6x  12x  54x C 24x C 180x C 156x C 216x C 72x C 18 12 9 8 6 5 4 3 2 T253 x  4x  3x  32x  48x  18x C 64x C 144x C 108x C 27 12 9 8 6 5 4 3 2 T252 x  12x C 27x C 12x  36x C 27x  16x C 36x C 9 12 6 5 4 3 2 T251 x C 48x  72x C 27x C 64x  144x C 108x  27 12 2 T250 x C 3x C 5 12 10 8 6 4 3 T249 x  12x C 54x  108x C 81x  8x C 24x C 8 12 6 5 4 3 2 T248 x C 324x  648x C 675x  744x C 648x  288x C 48 12 9 6 4 3 T247 x  8x C 24x C 162x  32x C 16 12 4 3 2 T246 x C 81x  216x C 216x  96x C 16 12 10 8 7 6 4 3 2 T245 x  12x  54x  72x C 96x C 9x C 200x C 108x  4 12 11 10 9 8 7 6 5 4 3 T244 x  3x  6x C 13x C 6x  15x C 5x  15x C 15x C 5x  5 12 8 7 6 4 3 2 T243 x  9x  12x  4x  81x  216x  216x  96x  16 12 9 8 6 5 4 3 2 T242 x  4x C 18x  4x  36x C 81x C 16x C 108x C 16 12 10 8 6 4 T241 x C x  3x  x C 6x  3 12 8 6 T240 x C 6x C 4x  4 12 9 8 6 5 4 3 2 T239 x  12x C 9x  32x C 48x  18x  64x C 144x  108x C 27 12 10 8 T238 x C 6x C 9x  8 12 2 T237 x C 10x C 5 12 4 2 T236 x C x C 2x C 1 12 8 6 T235 x C 3x  4x C 2 12 9 3 T234 x C x C 3x C 4 12 3 T233 x  4x  6 12 8 7 6 5 4 3 2 T232 x  13x  26x  11x C 6x C 25x C 78x C 114x C 76x C 19 12 11 9 8 7 6 5 4 3 2 T231 x  x C 2x  x  4x C 5x  x  x  x C 4x  3x C 1 12 10 8 4 T230 x C x  3x C 4x C 1 12 10 9 8 7 6 5 4 3 T229 x  18x  22x C 102x C 180x  96x  90x C 81x  30x 54x2 C 3 12 11 10 9 8 7 6 5 4 3 T228 x C 4x C 3x  2x C 11x C 30x C 14x  11x C 12x C 30x Cx2  9x  1 12 10 8 4 T227 x C 2x C x  4x C 3 12 8 2 T226 x  3x  6x C 1 12 10 6 4 T225 x  3x C 2x C 2x  3 12 8 6 2 T224 x C 4x C 6x  6x C 2 12 10 6 T223 x  6x C 12x  9 12 6 2 T222 x  4x C 3x  1 12 10 8 6 4 2 T221 x  2x  x C 6x  x  4x  1 508 Appendix: Example Polynomials

12 9 8 6 5 4 2 T220 x  4x  12x C 34x  12x C 45x C 42x C 10 12 6 2 T219 x C 2x C x C 1 12 11 9 7 5 3 T218 x  2x C 22x  88x C 176x  176x C 64x C 4 12 9 T217 x  4x C 2 12 10 9 4 3 2 T216 x  12x  8x C 162x C 432x C 432x C 192x C 32 12 10 9 8 7 6 5 4 3 T215 x  3x C x  81x C 54x  36x C 27x C 72x  107x C54x2  12x C 1 12 9 8 6 4 3 2 T214 x  12x C 18x  56x C 138x  96x C 72x C 72 12 3 T213 x  x  1 12 10 8 7 6 4 3 2 T212 x  12x  18x  96x  132x  63x  64x C 72x  16 12 8 7 6 4 3 2 T211 x C 90x C 120x C 40x C 405x C 1080x C 1080x C 480x C 80 12 9 6 5 4 3 2 T210 x  4x C 8x  36x C 105x  120x C 90x  36x C 9 12 9 8 7 6 4 3 T209 x  8x C 18x  24x C 24x  33x  16x  48x  8 12 10 6 4 T208 x  3x C 3x C 3x C 3 12 11 9 7 6 5 4 3 2 T207 x  x C x  x  x C 2x  x  3x C 3x  2x C 1 12 9 8 5 4 3 2 T206 x  12x C 15x  12x C 18x  64x C 96x  36x C 9 12 6 5 4 3 2 T205 x  208x  312x  117x  832x  1872x  1404x  351 12 9 8 6 4 3 T204 x  6x C 18x C 48x C 108x  32x  72x C 24 12 6 4 T203 x  2x C x C 1 12 6 4 T202 x  4x C 9x C 4 12 10 2 T201 x C 3x  12x C 24 12 10 8 T200 x C 6x C 9x  12 12 10 8 6 4 T199 x  2x  4x  x C x C 4 12 10 8 4 2 T198 x  2x  x C 6x  4x C 2 12 6 4 T197 x C 4x  9x C 8 12 8 6 4 2 T196 x  2x  4x C 6x C 4x  1 12 10 8 6 T195 x C 4x C 2x  4x C 4 12 10 9 8 7 6 5 4 3 2 T194 x C 2x C 2x  x  2x C 4x  12x C 6x C 2x C 18x C 27 12 6 4 T193 x C 6x C 6x C 3 12 10 8 6 4 2 T192 x  6x C x C 36x  30x  28x C 18 12 10 8 6 4 2 T191 x C x C 2x  x C 2x  3x C 1 12 10 8 6 4 2 T190 x C 2x  13x C 36x C 15x  38x  19 12 8 4 2 T189 x  6x C 12x C 13x C 5 12 10 6 2 T188 x  2x C 5x C 5x  1 12 6 2 T187 x C 8x  9x C 1 12 10 2 T186 x  x  x  1 12 4 T185 x  x  2 12 8 6 4 2 T184 x C x C 9x C 9x C 7x C 1 12 6 4 2 T183 x  7x  10x  5x C 1 12 9 8 6 5 4 3 2 T182 x  8x C 6x C 20x  24x C 18x  16x C 24x C 8 12 8 6 5 4 3 2 T181 x  18x  36x  72x C 54x  144x  216x  72 12 10 8 6 4 2 T180 x  2x C 5x  8x C 6x  4x C 1 12 11 10 9 8 7 6 5 4 3 L2.11/ x C x  8x  29x C 48x C 51x  5x C 275x C 642x C 208x C308x2 C 41x C 2 12 9 3 T178 x  x C 4x  1 12 9 3 T177 x  4x C 4x C 2 12 6 3 T176 x C 4x  8x C 8 12 11 10 9 7 6 5 2 T175 x  2x C 4x  2x C 4x  3x C 2x C x  2x C 1 12 10 8 6 4 3 2 T174 x C 12x C 54x C 20x  447x  384x  792x  1152x  368 3 Realizations over Q for Transitive Groups 509

12 8 7 6 4 2 T173 x  36x  48x  32x C 162x  288x C 128 12 10 9 7 6 4 3 2 T172 x C 12x  6x  54x  157x C 210x C 174x C 234x C 252x C 118 12 9 8 5 4 3 2 T171 x  8x  36x  72x C 81x C 64x  144x C 64 12 9 6 3 T170 x  x C 2x C 4x C 3 12 3 T169 x  8x C 18 12 6 3 T168 x  10x  12x  2 12 3 T167 x  3x C 3 12 10 8 6 4 3 2 T166 x C 18x C 135x C 348x C 63x  512x  270x C 729 12 9 8 3 2 T165 x  16x C 12x C 256x  576x C 432x  108 12 9 7 6 5 3 2 T164 x C 4x C 6x C 8x  54x C 88x  57x  90x C 111 12 8 6 4 2 T163 x  x  2x C x  2x C 1 12 8 6 4 2 T162 x  2x  8x C 14x  16x C 4 12 10 2 T161 x C 3x C 18x C 9 12 10 8 6 4 T160 x C x C x C x  4x C 5 12 10 8 6 4 2 T159 x C 4x  4x  24x  x C 32x C 8 12 8 6 2 T158 x  x  2x C 2x C 1 12 9 7 6 4 3 2 T157 x  8x C 24x C 44x  51x C 48x  72x C 16 12 9 T156 x  2x C 2 12 10 8 T155 x  2x  3x C 2 12 6 4 2 T154 x  2x C 12x  6x C 7 12 10 2 T153 x C 2x C 8x C 8 12 8 6 4 T152 x  4x  2x C 4x  1 12 8 T151 x  3x  2 12 6 4 2 T150 x  x  3x C 2x C 2 12 4 T149 x  9x  6 12 10 8 6 T148 x C 3x C 3x C x  3 12 8 T147 x  3x  8 12 10 8 6 4 2 T146 x  2x  x  2x  2x  8x C 8 12 8 6 4 2 T145 x C 6x C 4x  18x  24x  8 12 10 8 6 4 2 T144 x C 6x C 4x  24x  21x C 22x C 4 12 10 8 6 4 2 T143 x  6x C 24x  56x C 93x  90x C 51 12 8 6 4 T142 x C 3x C 4x C 6x C 3 12 8 T141 x C 3x  3 12 4 T140 x  x  4 12 10 2 T139 x C 3x C 3x C 1 12 4 T138 x  x C 1 12 8 6 4 T137 x C x  2x  x  1 12 10 2 T136 x  x C 4x C 1 12 8 6 4 2 T135 x  18x  24x C 27x C 36x  6 12 10 8 4 2 T134 x  7x C 14x  21x C 7x C 7 12 9 8 7 6 5 4 3 2 T133 x  8x C 162x  372x C 20x C 432x  63x  212x  36x C24x C 56 12 10 9 8 7 6 5 4 3 T132 x  x  11x C 99x  45x  117x  27x C 90x C 36x C 9x C 18 12 11 10 9 8 7 6 5 4 3 T131 x  2x  x C 9x  7x  11x C 20x C x  19x C 8x C6x2  5x C 1 12 9 6 3 T130 x  2x C x C 6x C 3 12 10 9 8 7 6 5 4 3 T129 x  6x  2x C 3x  30x C 8x C 90x C 36x  24x C 6x  1 12 10 9 8 6 5 4 3 T128 x  12x  22x C 57x  72x C 30x C 15x  30x C 6x C 1 12 9 8 6 5 4 3 T127 x  16x C 18x  72x C 36x  36x  76x  72x  62 12 8 6 4 2 T126 x C x C x  2x  x C 1 510 Appendix: Example Polynomials

12 8 6 4 2 T125 x  2x  2x C x C 2x  1 12 10 6 T124 x C 4x C 10x C 5 12 10 6 2 T123 x  2x C 10x  8x C 1 12 11 10 9 8 7 6 5 4 3 T122 x  2x  3x  6x C 21x  32x C 37x  16x C 11x C 32x x2 C 20x C 1 12 9 3 T121 x  x C 2x C 1 12 9 3 T120 x  2x  6x C 9 12 6 3 T119 x  8x  8x  2 12 6 3 T118 x C 8x  8x C 2 12 9 6 T117 x  2x C x C 5 12 9 3 T116 x  2x C 4x C 4 12 8 4 T115 x  2x C 3x  4 12 4 T114 x  x  1 12 4 T113 x  x C 4 12 8 4 T112 x  3x C 9x C 1 12 8 6 4 2 T111 x  6x C 68x C 105x C 36x C 12 12 8 6 4 T110 x C x  x  x  1 12 10 2 T109 x C x  4x C 1 12 8 6 4 T108 x  3x  4x C 6x C 4 12 10 8 6 4 2 T107 x C 6x C 3x  28x  21x C 30x C 5 12 10 8 6 2 T106 x C 3x  2x  9x C 5x C 1 12 10 8 6 4 2 T105 x  7x C 7x C 14x  16x  5x C 5 12 10 8 6 4 2 T104 x C 6x C 12x C 8x  3x  6x  1 12 10 6 2 T103 x C 3x  x C 3x C 1 12 10 8 6 4 2 T102 x  5x C 20x  70x C 145x  280x C 208 12 10 2 T101 x  3x  3x C 1 12 10 8 6 4 2 T100 x  x C x C 4x  x  x  1 12 8 6 4 T99 x  76x C 325x  380x C 125 12 10 8 6 4 T98 x  64x  231x C 740x  481x C 37 12 8 4 T97 x C x C 9x C 1 12 4 T96 x  3x  4 12 10 6 4 2 T95 x  x C 3x  2x  3x C 1 12 8 6 4 2 T94 x  57x  38x C 318x  204x C 17 12 10 8 6 4 2 T93 x C 10x C 28x C 6x  43x C 6x C 3 12 4 T92 x  9x  9 12 10 8 6 4 2 T91 x C 5x C 9x C 8x C 2x  12x C 16 12 10 6 2 T90 x C 2x  x C 2x C 1 12 4 T89 x  3x C 1 12 8 6 4 2 T88 x  6x  4x  3x  18x C 3 12 10 8 6 4 T87 x C 6x C 9x  4x  12x C 1 12 8 T86 x C 2x  2 12 11 10 9 8 7 6 5 4 3 T85 x  3x  3x C 15x  15x  33x C 29x C 15x  30x  128x 30x2 C 198x C 48 12 10 9 8 7 6 3 2 T84 x  6x C 4x C 21x  12x  52x  16x C 48x C 16 12 6 3 T83 x C 3x  x C 3 12 10 8 6 4 2 T82 x  12x C 54x  116x C 129x  72x  16 12 6 T81 x C x C 2 3 Realizations over Q for Transitive Groups 511

12 8 6 4 2 T80 x  90x C 160x  135x C 7200x  80 12 10 8 6 T79 x C 4x C 6x C 4x C 2 12 9 3 T78 x  x C x C 1 12 6 2 T77 x  2x C 5x C 1 12 8 4 2 T76 x C 2x C 5x C 6x C 1 12 8 4 2 T75 x C 7x C 7x C 8x C 1 12 10 8 6 4 2 T74 x  x C 2x C 4x  3x  3x C 1 12 11 10 8 7 6 5 4 3 2 T73 x  3x C 4x  x  6x C 20x  10x C 8x C 24x C 3x C 12x C 9 12 10 9 8 6 5 4 3 2 T72 x  6x  10x C 36x  116x C 720x C 696x  2440x  720x C1200x C 880 12 9 6 T71 x  4x C 4x C 3 12 6 3 T70 x C 9x  18x C 9 12 10 8 6 2 T69 x  3x  2x C 9x  5x C 1 12 10 8 6 4 2 T68 x C x C 6x C 3x C 6x C x C 1 12 8 6 4 T67 x  x  x  x C 1 12 10 8 6 T66 x C 6x C 12x C 8x  3 12 4 T65 x  3x C 4 12 8 T64 x C 3x  16 12 10 6 4 2 T63 x  6x C 104x C 93x C 18x C 4 12 10 8 6 4 2 T62 x  3x C 3x  x C 4x  4x C 1 12 4 T61 x  3x  1 12 8 4 T60 x  4x  9x C 4 12 10 8 6 4 T59 x  6x C 6x  4x  3x C 3 12 8 6 4 2 T58 x  12x  14x C 9x C 12x C 1 12 10 8 6 4 2 T57 x C 38x C 533x C 3474x C 10574x C 12740x C 4225 12 10 6 2 T56 x  2x C x  2x C 1 12 10 8 6 4 2 T55 x C 2x  97x  360x  345x  50x C 25 12 8 4 T54 x  6x C 9x C 2 12 8 6 4 T53 x C 2x  16x C 4x C 8 12 4 T52 x  3x  6 12 8 4 T51 x C 6x C 9x C 3 12 4 T50 x  3x C 6 12 8 6 4 T49 x C 3x  4x  3x  1 12 4 T48 x C 8x C 1 12 10 9 7 6 5 4 3 2 T47 x  6x C 20x  72x C 128x  96x C 45x  8x  18x C 12x  2 12 11 10 9 8 7 6 5 4 3 T46 x  4x C 6x C 4x  21x C 40x  28x  8x C 25x  28x C10x2  4x  1 12 9 8 6 5 4 3 T45 x  3x  18x  24x  9x C 69x  x C 3x  1 12 6 3 T44 x  6x  10x  6 12 9 6 3 T43 x  6x C 10x C 4x C 2 12 6 T42 x  x C 7 12 9 6 3 T41 x  x  6x C x C 1 12 10 8 6 4 2 T40 x  7x C 24x  36x C 24x C 13x C 1 12 6 T39 x  4x C 2 12 6 T38 x C x  3 12 6 T37 x C x C 4 12 9 3 T36 x  2x  2x C 1 12 9 6 3 T35 x  x  x C x C 1 512 Appendix: Example Polynomials

12 10 8 6 4 T34 x C 12x C 54x C 108x C 81x C 16 12 8 6 4 2 T33 x C 2x C 58x C 301x C 174x C 25 12 10 8 6 4 2 T32 x C 7x  x  23x  x C 7x C 1 12 10 8 6 4 2 T31 x C 6x  23x  210x  360x  50x C 25 12 10 8 6 4 2 T30 x  7x  14x C 115x  70x  175x C 125 12 8 6 4 2 T29 x  45x C 50x C 225x  375x C 125 12 T28 x C 2 12 10 8 6 4 2 T27 x C 12x C 68x C 220x C 392x C 360x C 148 12 8 6 4 T26 x  9x  8x  9x C 1 12 8 4 T25 x C 5x C 6x C 1 12 8 4 T24 x  2x  7x C 16 12 4 T23 x  4x C 4 12 10 8 7 6 5 4 2 T22 x  5x C 7x  6x  17x  6x C 7x  5x C 1 12 8 6 4 T21 x C 3x  4x C 3x C 1 12 9 8 7 6 5 4 3 2 T20 x  4x C 72x  84x C 236x  144x C 324x  192x C 72x C 8 12 10 8 6 4 2 T19 x C 24x C 196x C 600x C 452x C 112x C 8 12 6 T18 x C 2x C 4 12 8 6 4 2 T17 x C 4x C 4x C 5x C 12x C 2 12 6 T16 x  x C 4 12 T15 x C 3 12 6 T14 x  9x C 27 12 T13 x  3 12 6 T12 x C x  27 12 6 T11 x  8x C 8 12 T10 x C 9 12 8 6 4 T9 x C 3x C 4x C 3x C 1 12 10 9 8 7 6 4 3 T8 x  6x  8x C 9x C 12x  20x C 9x  24x  4 12 10 8 4 2 T7 x C 4x  x  x C 4x C 1 12 10 8 6 4 2 T6 x C 2x  6x C 2x  6x C 2x C 1 12 10 8 6 4 2 T5 x  80x C 1820x  13680x C 29860x  2720x C 32 12 8 6 4 2 T4 x C 6x C 26x  63x C 162x C 81 12 T3 x C 36 12 6 T2 x  x C 1 12 11 10 9 8 7 6 5 4 3 2 T1 x  x C x  x C x  x C x  x C x  x C x  x C 1

Table 3.12 Degree 13 13 T9 S13 x  x C 1 13 T8 A13 x C 156x  144 13 12 10 9 8 7 6 5 4 T7 L3.3/ x C x C 40x C 13x  99x C 180x  468x  468x C 1644x 912 C 24x C 24 13 T6 F156 x  2 13 9 8 7 6 5 4 3 2 T5 F78 x C 3x  10x  3x C 5x  20x  11x C 2x  10x  10x  3 13 10 8 7 6 4 2 T4 F52 x C 13x  26x C 13x C 52x  39x C 26x C 13x C 2 13 11 9 7 6 5 4 3 T3 F39 x  39x C 468x  1989x  507x C 2886x C 1443x  624x 234x2 C 3 13 12 10 9 8 7 6 5 4 3 T2 D13 x  2x C 4x  5x C x C 5x  11x C 19x  22x C 16x 10x2 C 6x  1 13 12 11 10 9 8 7 6 5 T1 13 x  x  24x C 19x C 190x  116x  601x C 246x C 738x 215x4  291x3 C 68x2 C 10x  1 3 Realizations over Q for Transitive Groups 513

Table 3.13 Degree 14 14 S14 x  x  1 14 7 5 A14 x  9x C 49x  90 14 2 T61 x C x  2x C 1 14 8 7 2 T60 x  7x  6x C 49x C 84x C 36 14 7 2 T59 x  96x  1568x C 2304 14 8 7 2 T58 x  7x C 6x C 784x  1344x C 576 14 2 T57 x  x C 1 14 8 T56 x C 14x  24 14 2 T55 x  x  1 14 6 T54 x C 7x C 4 14 8 6 T53 x C 7x  7x  9 14 13 12 11 10 8 7 5 4 3 2 T52 x  x C x  x C x  4x C 6x  5x C 5x C x  4x C 1 14 2 T51 x  7x  3 14 8 6 2 T50 x  2x  5x  3x  4 14 6 T49 x  4x C 4 14 6 2 T48 x C 7x C 21x C 50 14 12 10 6 4 2 T47 x C 2x  2x C x  8x C 5x C 2 14 10 8 T46 x C 5x  4x C 2 14 12 11 10 9 8 7 6 5 4 T45 x  7x  14x C 21x C 84x C 35x  69x C 7x C 84x C 7x C77x3 C 133x2 C 35x C 58 14 10 8 6 4 2 T44 x  8x  2x C 16x C 6x  6x  2 14 12 8 6 2 T43 x C 3x  4x C x  3x C 1 14 12 10 8 6 4 2 T42 x C 7x  7x  49x C 7x C 49x  49x C 9 14 12 10 8 6 2 T41 x  2x  2x C x C 6x  x  4 14 12 8 6 4 2 T40 x C 2x  14x C 35x  21x  7x C 7 14 13 10 6 5 2 PGL2.13/ x  x  26x C 65x C 13x  52x  12x  1 14 8 6 T38 x  7x  14x  7 14 11 9 8 7 6 4 3 T37 x  28x  28x C 196x  2x C 392x C 616x  392x C14x2 C 56x C 9 14 12 11 10 9 8 7 6 T36 x  35x  133x C 469x C 1239x C 742x  3604x C 47138x 85351x5 C 168028x4  156394x3 C 158718x2  72149x C 42751 14 12 10 8 6 4 2 T35 x  9x C 17x C 29x  49x  67x  21x  1 14 12 8 6 2 T34 x  3x C 4x C x  3x  1 14 10 8 6 4 2 T33 x C 14x C 28x  35x C 784x  140x  4 14 12 10 8 7 6 5 4 3 T32 x  14x C 77x  210x C x C 294x  7x  196x C 14x C49x2  7x C 2 14 12 8 7 5 4 3 2 T31 x  7x C 91x  192x  126x  1519x C 1218x C 8827x C11046x C 5484 14 13 12 9 8 4 L2.13/ x  6x C 13x  338x C 845x C 17576x C 70304x C 35152 14 12 10 8 6 4 2 T29 x C 12x C 41x C 26x  59x  64x C 9x C 17 14 6 4 2 T28 x C 7x C 7x C 7x  1 14 8 6 T27 x C 7x  14x C 7 14 11 10 9 8 7 6 5 T26 x  28x C 280x C 567x C 5061x C 2273x  735x C 33908x C40348x4  3192x3 C 36855x2 C 119196x C 75141 514 Appendix: Example Polynomials

14 12 11 10 9 8 7 6 T25 x C 42x  42x C 525x  896x C 2422x  2536x C 1225x C742x5  994x4 C 560x3  28x2  168x C 56 14 7 T24 x  3x C 6 14 12 10 8 7 6 5 4 3 2 T23 x  14x C 77x  210x  11x C 294x C 77x  196x  154x C 49x C77x C 29 14 12 11 10 9 8 7 T22 x C 42x  840x C 4473x  77728x C 235648x  2601696x C6832756x6  48638016x5 C 124211584x4  490172256x3 C802837840x2  1497646080x C 723639232 14 12 10 8 6 4 2 T21 x  x  12x C 7x C 28x  14x  9x  1 14 13 12 11 9 8 7 6 4 3 2 T20 x  2x  4x C x C 6x C 10x  x C 6x  13x  15x  5x C x  1 14 8 6 4 T19 x C 10x C 8x  4x C 2 14 12 10 8 6 4 2 T18 x C 4x  30x C 8x C 60x C 8x  24x  8 14 12 10 8 6 4 2 T17 x C 11x C 53x C 15x  149x C 89x  x  3 14 10 8 7 6 4 3 2 T16 x  14x C 14x C 22x C 21x C 49x  154x C 77x  154x C 149 14 12 10 9 8 7 6 5 T15 x  87x C 1456x  256x  8563x C 3448x C 18032x  9890x 11776x4 C 5198x3 C 3128x2  506x  184 14 7 T14 x  2x C 8 14 13 11 10 9 8 7 6 5 4 T13 x C 4x C 10x C 39x C 28x  13x C 34x C 126x  36x C 29x 24x3 C 38x2  16x C 4 14 12 11 10 9 8 7 6 T12 x C 35x C 210x C 735x C 2849x C 10150x C 45655x C 94570x C98455x5  199381x4  344400x3 C 647395x2 C 4094650x C 1010645 14 12 10 8 6 4 2 T11 x  5x  11x C 25x C 27x  23x  17x  1 14 8 6 4 2 T10 x C 14x  84x C 84x C 21x  9 14 12 10 8 6 4 T9 x C 7x  49x  245x C 588x C 294x  7 14 12 11 10 9 8 7 6 5 4 3 2 T8 x  x  3x C 5x C 5x  5x  9x C x C 14x  2x  7x C x C 1 14 T7 x C 2 14 12 10 8 6 4 2 T6 x C 13x C 31x  9x  54x  3x C 23x  1 14 7 T5 x  x C 2 14 T4 x C 7 14 12 10 8 6 4 2 T3 x C 6x C 7x C x  3x C x C 3x C 1 14 12 10 8 6 4 2 T2 x C 8x C 22x C 8x  55x  48x C 64x C 71 14 12 10 8 6 4 2 T1 x C 25x C 214x C 767x C 1194x C 686x C 53x C 1 References

Abhyankar, S.S. (1957): Coverings of algebraic curves. Amer. J. Math. 79, 825–856 Abhyankar, S.S. (1994): Nice equations for nice groups. Israel J. Math. 88, 1–23 Abhyankar, S.S. (1996a): Again nice equations for nice groups. Proc. Amer. Math. Soc. 124, 2967–2976 Abhyankar, S.S. (1996b): More nice equations for nice groups. Proc. Amer. Math. Soc. 124, 2977–2991 Abhyankar, S.S., Inglis, N.J. (2001): Galois groups of some vectorial polynomials. Trans. Amer. Math. Soc. 353, 2941–2869 Abhyankar, S.S., Keskar, P.H. (2001): Descent principle in modular Galois theory. Proc. Indian Acad. Sci. Math. 11, 139–149 Abhyankar, S.S., Loomis, P.A. (1998): Once more nice equations for nice groups. Proc. Amer. Math. Soc. 126, 1885–1896 Abhyankar, S.S., Loomis, P.A. (1999): Twice more nice equations for nice groups. Pp. 63–76 in: Applications of curves over finite fields (Seattle, WA, 1997), Con- temp. Math., 245, Amer. Math. Soc., Providence, RI, 1999 Albert, M., Maier, A. (2011): Additive polynomials for finite groups of Lie type. Israel J. Math. 186, 125–195 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 Barth, D., Wenz, A. (2016): Explicit Belyi maps over Q having almost simple prim- itive monodromy groups. Preprint, arXiv:1703.02848

© Springer-Verlag GmbH Germany, part of Springer Nature 2018 515 G. Malle, B. H. Matzat, Inverse Galois Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-662-55420-3 516 References

Barth, D., Wenz, A. (2017): Belyi map for the sporadic group J1. Preprint, arXiv:1704.06419 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), 247–256] 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 D 6;7;8). J. Algebra 88, 584–614 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., G¨untzer, U., Remmert, R. (1984): Non-archimedean analysis. Springer, Berlin Heidelberg New York Breuer, T., Malle, G., O’Brien, E. (2016): Reliability and reproducibility of Atlas information. Submitted 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 Carter, R.W. (1985): Finite groups of Lie type: Conjugacy classes and complex char- acters. John Wiley and Sons, Chichester Carter, R.W. (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 Colin, A. (1995): Formal computation of Galois groups with relative resolvents. Pp. 169–182 in: Lecture Notes in Comput. Sci., 948, Springer, Berlin References 517

Conway, J.H. et al. (1985): Atlas of finite groups. Clarendon Press, Oxford n Cooperstein, B.N. (1981): Maximal subgroups of G2.2 /.J.Algebra70, 23–36 Q 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 of reductive 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,ˇ S.P., Safareviˇ c,ˇ 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)] Dentzer, R. (1989): Projektive symplektische Gruppen PSp4.p/ als Galoisgruppen uber¨ 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 E8. Tokyo J. Math. 6, 191–216 Deriziotis, D.I., Michler, G.O. (1987): Character table and blocks of finite simple 3 triality groups D4.q/.Trans.Amer.Math.Soc.303, 39–70 Derksen, H., Kemper, G. (2002): Computational invariant theory. Springer Verlag, Berlin Dettweiler, M. (2003): Middle convolution functor and Galois realizations. Pp. 143– 158 in: K. Hashimoto et al. Eds: Galois theory and modular forms. Kluwer Dettweiler, M. (2004): Plane curve complements and curves on Hurwitz spaces. J. reine angew. Math. 573, 19–43 Dettweiler, M., Reiter, S. (1999): On rigid tuples in linear groups. J. Algebra 222, 550–560 Dettweiler, M., Reiter, S. (2000): An algorithm of Katz and its applications to the inverse Galois problem. J. Symb. Comput. 30, 761–798 Dew, E. (1992): Fields of moduli of arithmetic Galois groups. PhD-thesis, Univer- sity 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 reductifs´ 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 Elkies, N.D. (1997): Linearized algebra and finite groups of Lie type I: Linear and symmetric groups. Contemp. Math. 245, 77–108 Engler, A.J., Prestel, A. (2005): Valued fields. Springer Verlag, Berlin 518 References

f Enomoto, H. (1976): The characters of the finite Chevalley groups G2.q/, q D 3 . Japan. J. Math. 2, 191–248 n Enomoto, H., Yamada, H. (1986): The characters of G2.2 /. 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, 190–243 (Russian) [English transl.: Amer. Math.Soc.Transl.II3, 15–38 (1956)] 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 S 2. Duke Math. J. 29, 243–257 2 Feit, W. (1984): Rigidity of Aut.PSL2.p //, p Á˙2.mod 5/, p ¤ 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 Galoisgruppenuber ¨ 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): Geom´ etrie´ analytique rigide et applications. Birkhauser,¨ Basel Fresnel, J., van der Put, M. (2004): Rigid analytic geometry and its applications. Birkhauser,¨ Boston 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, Amer. Math. Soc., Providence References 519

Frohlich,¨ A. (1985): Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants. J. reine angew. Math. 360, 84–123 Frohardt, D., Guralnick, R., Hoffman, C., Magaard, K. (2016): Exceptional low genus actions of finite groups. Preprint Frohardt, D., Magaard, K. (2001): Composition factors of monodromy groups. Ann. of Math. 154, 327–345 Garcia Lopez, F. (2010): An algorithm for computing Galois groups of additive polynomials. Preprint Geck, M. (2003): An introduction to algebraic geometry and algebraic groups. Oxford University Press, Oxford Geyer, W.-D., Jarden, M. (1998): Bounded realization of `-groups over global fields. Nagoya Math. J. 150, 13–62 Gille, P. (2000): Le groupe fondamental sauvage d’une courbe affine in character- istic p. Pp. 217–231 in: Bost, J.-L. et al. eds.: Courbes semi-stables et groupe fondamental en geometrie´ algebrique.´ Birkhauser,¨ Basel Gillette, 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 reguli´ ere` de Q.T / de groupe de Galois M24. Experimental Math. 5, 3–14 Grothendieck, A. (1971): Revetementsˆ etales´ et groupe fondamental. Springer, Berlin Guralnick, R.M., L¨ubeck, F., Yu, J. (2016): Rational rigidity for F4.p/.Adv.Math. 302, 48–58 Guralnick, R.M., Malle, G. (2012): Simple groups admit Beauville structures. J. London Math. Soc. 85, 694–721 Guralnick, R.M., Malle, G. (2014): Rational rigidity for E8.p/. Compos. Math. 150, 1679–1702 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- penuber ¨ Q. Math. Ann. 292, 587–618 Hansen, V.L.(1989): Braids and coverings. Cambridge University Press, Cambridge Haraoka, Y. (1994): Finite monodromy of Pochhammer equation. Ann. Inst. Fourier 44, 767–810 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 Harbater, D. (1994a): Abhyankar’s conjecture on Galois groups over curves. Invent. Math. 117, 1–25 520 References

Harbater, D. (1994b): Galois groups with prescribed ramification. Pp. 35–60 in: N. Childress and J. Jones (eds.): Arithmetic Geometry. Contemporary Math. vol. 174, Amer. Math. Soc., Providence Harbater, D. (1995a): Fundamental groups and embedding problems in characteris- tic 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., Z¨urich 1994 , Birkhauser¨ Hartshorne, R. (1977): Algebraic geometry. Springer, Berlin Heidelberg New York Hasse, H. (1948): Existenz und Mannigfaltigkeit abelscher Algebren mit vorgege- bener Galoisgruppeuber ¨ 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 uni- tary groups. J. reine angew. Math. 218, 174–189 Hilbert, D. (1892): Uber¨ 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 J1 und J2 als Galoisgruppenuber ¨ Q. J. Algebra 97, 14–22 Hoyden-Siedersleben, G., Matzat, B.H. (1986): Realisierung sporadischer einfacher Gruppen als Galoisgruppenuber ¨ 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): Uber¨ Riemann’sche Flachen¨ mit gegebenen Verzweigungs- punkten. Math. Ann. 39, 1–61 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 al. (eds.) (1989): Galois groups over Q. Springer, New York Iitaka, S. (1982): Algebraic geometry. Springer Verlag, Berlin Ikeda, M. (1960): Zur Existenz eigentlicher galoisscher Korper¨ beim Einbettungs- problem f¨ur galoissche Algebren. Abh. Math. Sem. Univ. Hamburg 24, 126–131 Inaba, E. (1944): Uber¨ 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. Amer. Math. Soc., Providence Iwasawa, K. (1953): On solvable extensions of algebraic number fields. Ann. of Math. 58, 548–572 References 521

Jacobson, N. (1980): Basic algebra II. W. H. Freeman, New York Jensen, C., Ledet, A., Yui, N. (2002): Generic polynomials. Constructive aspects of the inverse Galois problem. Cambridge University Press, Cambridge Kantor, W.M. (1979): Subgroups of classical groups generated by long root ele- ments. Trans. Amer. Math. Soc. 248, 347–379 Katz, N. (1996): Rigid local systems. Annals of Math. Studies 139, Princeton Uni- versity Press, Princeton Kemper, G., L¨ubeck, F., Magaard, K. (2001): Matrix generators for the Ree groups 2 G2.q/. Comm. Algebra 29, 407–415 Kleidman, P.B. (1987): The maximal subgroups of the finite 8-dimensional orthog- C onal groups P8 .q/ and of their automorphism groups. J. Algebra 110, 173–242 Kleidman, P.B. (1988a): The maximal subgroups of the Steinberg triality groups 3 D4.q/ and of their automorphism groups. J. Algebra 115, 182–199 Kleidman, P.B. (1988b): The maximal subgroups of the Chevalley groups G2.q/ 2 with q odd, the Ree groups G2.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 Klein, F. (1884): Vorlesungenuber ¨ das Ikosaeder. Teubner, Leipzig Klein, F., Fricke, R. (1890): Vorlesungenuber ¨ die Theorie der elliptischen Modul- funktionen I. Teubner, Leipzig Kl¨uners, J., Malle, G. (2000): Explicit Galois realization of transitive groups of degree up to 15. J. Symbolic Comput. 30, 675–716 Kl¨uners, J., Malle, G. (2002): A Database for Number Fields. http://galoisdb.math. upb.de/ Kochendorffer,¨ R. (1953): Zwei Reduktionssatze¨ zum Einbettungsproblem f¨ur abel- sche Algebren. Math. Nachr. 10, 75–84 Konig,¨ J. (2015): Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups. Math. Comp., to appear Konig,¨ J. (2016): On rational functions with monodromy group M11. J. Symbolic Comput. 79, 372–383 Kopf,¨ U. (1974): Uber¨ eigentliche Familien algebraischer Varietaten¨ uber ¨ affinoiden Raumen.¨ Schriftenreihe Math. Inst. Univ. M¨unster, 2. Serie, Heft 7 Kreuzer, M., Robbiano, L. (2000): Computational commutative algebra I. Springer Verlag, Berlin Krull, W., Neukirch, J. (1971): Die Struktur der absoluten Galoisgruppeuber ¨ dem Korper¨ IR.t/. Math. Ann. 193, 197–209 Kucera, J. (1994): Uber¨ die Brauergruppe von Laurentreihen- und rationalen Funk- tionenkorpern¨ und deren Dualitat¨ mit K-Gruppen. Dissertation, Universitat¨ 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 representa- tions of the finite Chevalley groups. J. Algebra 32, 418–443 Lang, S. (1982): Introduction to algebraic and abelian functions. Springer, New York 522 References

Ledet, A. (2000): On a theorem of Serre. Proc. Amer. Math. Soc. 128, 27–29 Ledet, A. (2005): Brauer type embedding problems. Amer. Math. Soc., Providence Liebeck, M.W., Seitz, G.M. (1999): On finite subgroups of exceptional algebraic groups. J. Reine Angew. Math. 515, 25–72 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) L¨ubeck, F., Malle, G. (1998): .2;3/-generation of exceptional groups. J. London Math. Soc. 59, 109–122 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, Universitat¨ Karlsruhe Malle, G. (1987): Polynomials for primitive nonsolvable permutation groups of degree d Ä 15.J.Symb.Comp.4, 83–92 Malle, G. (1988a): Polynomials with Galois groups Aut.M22/,M22,and PSL3.IF 4/:2 over Q. Math. Comp. 51, 761–768 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/ uber¨ 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. (2003): Explicit realization of the Dickson groups G2.q/ as Galois groups. Pacific J. Math. 212, 157–167 Malle, G., Matzat, B. H. (1985): Realisierung von Gruppen PSL2.IFp/ als Galois- gruppenuber ¨ Q. Math. Ann. 272, 549–565 Malle, G., Matzat, B. H. (1999): Inverse Galois theory. Springer Verlag, Berlin– Heidelberg Malle, G., Saxl, J., Weigel, T. (1994): Generation of classical groups. Geom. Dedi- cata 49, 85–116 References 523

Malle, G., Sonn, J. (1996): Covering groups of almost simple groups as Galois groups over Qab.t/. Israel J. Math. 96, 431–444. Malle, G., Testerman, D. (2011): Linear algebraic groups and finite groups of Lie type. Cambridge University Press, Cambridge Matsumura, H. (1980): Commutative algebra. Benjamin/Cummings, Reading Matzat,p B.H. (1979): Konstruktion von Zahlkorpern¨ mit der Galoisgruppe M11 uber¨ Q. 11/. 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-Einbettungsproblemeuber ¨ 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. 74, 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. (2003): Frobenius modules and Galois groups. Pp. 233-268 in K. Hashimoto, K. Miyake, H. Nakamura et al. Eds.: Galois theory and modular forms. Kluwer Acad. Publ., Boston, MA Matzat, B.H., Zeh-Marschke, A. (1986): Realisierung der Mathieugruppen M11 und M12 als Galoisgruppenuber ¨ Q. J. Number Theory 23, 195–202 McLaughlin, J. (1969): Some subgroups of SLn.IFp/. Ill. J. Math. 13, 108–115 Melnicov, O.V. (1980): Projective limits of free profinite groups. Dokl. Akad. Nauk SSSR 24, 968–970 Q Mestre, J.-F. (1990): Extensions reguli´ eres` de Q.T / de groupe de Galois An.J. Algebra 131, 483–495 Mestre, J.-F. (1994): Construction d’extensions reguli´ eres` de Q.t/ a` groupes de Q Galois SL2.F7/ et M12.C.R.Acad.Sci.319, 781–782 Mestre, J.-F. (1998): Extensions Q-reguli´ eres` de Q.t/ de groups de Galois 6:A6 et 6:A7. Israel J. Math. 107, 333–341 524 References

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 Monien, H. (2017): The sporadic group J2, Hauptmodul and Belyi map. Preprint, arXiv:1703.05200 M¨uller, P. (2012): A one-parameter family of polynomials with Galois group M24 over Q.t/. Preprint, arXiv:1204.1328 Nagata, M. (1977): Field theory. Marcel Dekker, New York Nakamura, H. (1997): Galois rigidity of profinite fundamental groups. Sugaku Expositiones 10, 195–215 Narkiewicz, W. (1990): Elementary and analytic theory of algebraic numbers. Springer, Berlin Neukirch, J., Schmidt, A. and Wingberg, K. (2000): Cohomology of Number Fields. Springer-Verlag, 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 Nori, M.V. (1994): Unramified coverings of the affine line in positive characteris- tic. Pp. 209–212 in: Bajaj, C.L., ed.: Algebraic geometry and its applications. Springer, New York n Norton, S.P., Wilson, R.A. (1989): The maximal subgroups of F4.2 / 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] Pochhammer, L. (1870): Uber¨ die hypergeometrischen Funktionen n-ter Ordnung. J. reine angew. Math. 71, 312–352 1 Pop, F. (1994): 2 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 n Porsch, U. (1993): Greenfunktionen der endlichen Gruppen E6.q/, q D 3 . Diplom- arbeit, Universitat¨ Heidelberg Przywara, B. (1991): Zopfbahnen und Galoisgruppen. IWR-Preprint 91-01, Univer- sitat¨ Heidelberg van der Put, M., Singer, M.F. (1997): Galois theory of difference equations. Springer Verlag, Berlin References 525

Raynaud, M. (1994): Revetementsˆ de la droite affine en caracteristique´ p>0et 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. (1999): Galoisrealisierungen klassischer Gruppen. J. reine angew. Math. 511, 193–236 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 Safareviˇ c,ˇ I.R. (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)] Safareviˇ c,ˇ I.R. (1954a): On the construction of fields with a given Galois group of order `a. Izv. Akad. Nauk. SSSR 18, 216–296 (Russian) [English transl.: Amer. Math.Soc.Transl.II4, 107–142 (1956)] Safareviˇ c,ˇ I.R. (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)] Safareviˇ c,ˇ 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)] Safareviˇ c,ˇ I.R. (1954d): Construction of fields of algebraic numbers with given solvable Galois group. Izv. Akad. Nauk. SSSR 18, 525–578 (Russian) [English transl.: Amer. Math. Soc. Transl. II 4, 185–237 (1956)] Safareviˇ c,ˇ I.R. (1958): The embedding problem for splitting extensions. Dokl. Akad. Nauk SSSR 120, 1217–1219 (Russian) Safareviˇ c,ˇ I.R. (1989): Factors of descending central series. Math. Notes 45, 262– 264 Safareviˇ c,ˇ I.R. (1994): Basic algebraic geometry I. Springer Verlag Sa¨ıdi, M. (2000): Abhyankar’s conjecture II: the use of semi-stable curves. Pp. 249– 265 in: Bost, J.-L. et al. eds.: Courbes semi-stables et groupe fondamental en geometrie´ algebrique.´ Birkhauser,¨ Basel 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): Uber¨ 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 Q 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): Uber¨ die Bildung algebraischer Zahlkorper¨ mit auflosbarer¨ galoisscher Gruppe. Math. Z. 30, 332–356 Scholz, A. (1937): Konstruktion algebraischer Zahlkorper¨ mit beliebiger Gruppe von Primzahlpotenzordnung I. Math. Z. 42, 161-188 526 References

Scott, L. (1977): Matrices and cohomology. Ann. of Math. 105, 473–492 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): Geom´ etrie´ algebrique´ et geom´ etrie´ 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 revetementsˆ 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 2 Shiina, Y. (2003a): Braid orbits related to PSL2.p / and some simple groups. Tohoku Math. J. 55, 271–282 2 Shiina, Y. (2003b): Regular Galois realizations of PSL2.p / over Q.T /. Pp. 125– 142 in: K. Hashimoto et al. Eds: Galois theory and modular forms. Kluwer 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 ¤ 2. J. Fac. Sci. Univ. Tokyo 21, 1–17 Shoji, T. (1982): On the Green polynomials of a Chevalley group of type F4. Comm. Algebra 10, 505–543 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. (2000): Some polynomials over Q.t/ and their Galois groups. Math. Comp. 69, 775–796 Smith, L. (1995): Polynomial invariants of finite groups. A.K. Peters, Wellesley Sonn, J. (1990): On Brauer groups and embedding problems over function fields. J. Algebra 131, 631–640 References 527

Sonn, J. (1991): Central extensions of Sn as Galois groups of regular extensions of Q.T /.J.Algebra140, 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 Qab.t/. J. Number Theory 47, 398–404 3 Spaltenstein, N. (1982): Caracteres` unipotents de D4.IF q/. Comment. Math. Hel- vetici 57, 676–691 Speiser, A. (1919): Zahlentheoretische Satze¨ aus der Gruppentheorie. Math. Z. 5, 1–6 Springer, T.A. (1998): Linear algebraic groups. Birkhauser,¨ Basel Steinberg, R. (1965): Regular elements of semi-simple algebraic groups. Publ. Math. IHES 25, 49–80. Steinberg, R. (1967): Lectures on Chevalley groups. Yale University Steinberg, R. (1968): Endomorphisms of linear algebraic groups. Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence Stichel, D. (2014): Finite groups of Lie type as Galois groups over IFq .t/.J.Com- mut. Algebra 6, 587–603 Stocker,¨ R., Zieschang, H. (1988): Algebraische Topologie. Teubner, Stuttgart Stoll, M. (1995): Construction of semiabelian Galois extensions. Glasgow Math. J. 37, 99–104 Strammbach, K., Volklein,¨ H. (1999): On linearly rigid tuples. J. reine angew. Math. 510, 57–62 Stroth, G. (1998): Algebra. de Gruyter Verlag, Berlin Suzuki, M. (1962): On a class of doubly transitive groups. Ann. of Math. 75, 105–145 Suzuki, M. (1982): Group theory. Springer, Berlin Q 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 Ä Q.n/.J.Algebra89, 437–499 Thompson, J.G. (1984b): Primitive roots and rigidity. Pp. 327–350 in: M. Aschbacher et al. (eds.) (1984) Thompson, J.G. (1986): Regular Galois extensions of Q.x/, Pp. 210–220 in: Tuan Hsio-Fu (ed.): Group theory, Beijing 1984. Springer, Berlin Thompson, J.G., Volklein,¨ H. (1998): Symplectic groups as Galois groups. J. Group Theory 1, 1–58 528 References

Tschebotarow,¨ N.G., Schwerdtfeger, H. (1950): Grundz¨uge der Galoisschen Theo- rie. 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 2 PGLn.q/ and PUn.q /. Forum Math. 6, 513–535 Volklein,¨ H. (1996): Groups as Galois groups. Cambridge University Press, Cam- bridge Volklein,¨ H. (1998): Rigid generators of classical groups. Math. Ann. 311, 421–438 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. Trans. Amer. Math. Soc. 121, 62–89 Washington, L. (1982): Introduction to cyclotomic fields. Springer, New York Wehrfritz, B.A.F. (1973): Infinite linear groups. Springer Verlag, Berlin 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. (2017): Maximal subgroups of sporadic groups. To appear in: Finite Simple Groups: Thirty Years of the Atlas and Beyond, AMS Contemporary Math- ematics Series, 2017. 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 transl.: Math. USSR Izv. 1, 195–208 (1967)] Zalesskiˇı, A.E., Sereˇzkin, V.N. (1980): Finite linear groups generated by reflec- tions. Izv. Akad. Nauk SSSR Ser. Math. 44, 1279–1307 (Russian) [English transl.: Math. USSR Izv. 17 (1981), 477–503] References 529

Zassenhaus, H. (1958): Gruppentheorie. Vandenhoeck & Ruprecht, Gottingen¨ Zywina, D (2015): The inverse Galois problem for PSL2.IFp/. Duke Math. J. 164, 2253–2292. Index

-equivalent matrix, 386 conformal orthogonal group, 111 conformal symplectic group, 108 accompanying Brauer embedding problem, connected rigid analytic space, 452 350 convolution, 251 accompanying embedding problem, 350 coroot, 93 admissible covering, 451 cyclic F-module, 385 admissible subset, 451 cyclotomic character, 14 affinoid analytic space, 451 cyclotomic polynomial, 117 algebraic fundamental group, 4, 187 almost character, 127 Dedekind criterion, 72 arithmetic fundamental group, 10, 197 Dickson algebra, 396 (full) Artin braid group, 179 Dickson invariants, 395 associated F-module, 390 Dickson polynomial, 395 AV -rigid, 64 disclosed function field of one variable, 14 AV -symmetric, 64 duality theorem of Tate, 354 AV -symmetrized irrationality degree, 64 dualizable F-module, 385 effective G-module, 392 basic rigidity theorem, 30 embedding problem, 288 Belyi triple, 102 existentially closed, 474 braid cycle theorem, 246 extension theorem, 67 braid orbit theorem, 215 braid relations, 181 F-field, 385 Brauer embedding problem, 339 F-module, 385 field of definition, 19 central embedding problem, 288 field of definition with group, 19 characteristic polynomial of an F-module, 389 field of invariants, 394 clean Belyi function, 17 field of moduli, 30 closed ultrametric disc, 462 field restriction of algebraic groups, 441 coherent sheaf, 454 finite embedding problem, 288 cohomologically trivial in dimension i, 361 finite morphism, 458 companion matrix, 390 first embedding obstruction, 364 comparison theorem of Tate, 367 fixed point theorem, 53 compatible family, 467 Frattini embedding problem, 288 concordance obstruction, 364 Frattini embedding theorem, 319 concordant embedding problem, 351 Freiheitssatz of Iwasawa, 295

© Springer-Verlag GmbH Germany, part of Springer Nature 2018 531 G. Malle, B. H. Matzat, Inverse Galois Theory, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-662-55420-3 532 Index

Frobenius endomorphism, 385 large field, 475 Frobenius field, 385 Lemma of Scott, 260 Frobenius module, 385 Lemma of Speiser, 201 full symmetry group, 31, 63 linear Tschirnhaus transform, 401 fundamental solution matrix, 387 linearly rigid tuple, 260 fundamental system of solutions, 387 Lusztig series, 126

G-compatible family, 469 M-section, 469 G-realization, 34 mapping class group, 183 G-relative H-invariant, 396 modular Dedekind criterion, 403 G-relative Colin Matrix, 399 modular Galois theory, 383 G-relative resolvent, 396 Moore determinant, 387 GA-realization, 36 Moore matrix, 387 GAGA for IP1, 456 morphism of rigid analytic spaces, 451 Galois group of an F-module, 390 multiplication with c, 267 GAR-realization, 302 general unitary group, 107 non-split embedding problem, 288 generating s-system, 26 normalized structure constant, 36 generic polynomial, 396 geometric (proper) solution, 289 open ultrametric disc, 462 geometric embedding problem, 289 orthogonal group, 110 geometric field extension, 8 orthogonal group of minus type, 115 geometrically conjugate, 126 orthogonal group of plus type, 111 GL-stable tuple, 260 } gluing datum, 451 -stable, 86 gluing of morphisms, 452 pairwise adjusted, 471 gluing of spaces, 451 Pochhammer transform, 251 good reduction modulo p,88 Pochhammer transformation, 251 Green function, 128 primitive linear group, 100 group of geometric automorphisms, 43 primitive prime divisor, 118 primitive translate, 55 V profinite Hurwitz braid group, 189 H -rigid class vector, 212 s profinite Riemann existence theorem, 4 Hasse embedding obstruction, 364 projective profinite group, 294 Hasse-Witt-invariant, 332 proper solution (field) of an embedding Hilbertian field, 287 problem, 288 Hilbertian set, 287 pseudo algebraically closed, 229 homology, 100 pseudo Steinberg cross section, 424 homomorphism ramified in, 480 pseudo-reflection, 100 (full) Hurwitz braid group, 181 pure Artin braid group, 179 Hurwitz classification, 27, 198 pure Hurwitz braid group, 181 hypothesis (H), 254 q-additive polynomial, 388 induced cover, 461 quasi-central element, 132 irrationality degree, 28 quasi-determinant, 115 irreducible Jordan–Pochhammer tuple, 270 quasi-p-group, 484 j -th braid orbit genus, 213 r-fold uncomplete product, 179 Jordan–Pochhammer tuple, 270 r-fold uncomplete symmetric product, 179 rational class vector, 29 k-rational class vector, 319 rational subset, 450 k-symmetric class vector, 319 rationally rigid class vector, 29 kernel of an embedding problem, 288 reduced braid orbit genera, 245 Index 533 reflection, 100 Steinberg cross section, 406 regular solution of an embedding problem, 289 Steinberg endomorphism, 423 regularity theorem, 212 strictly non-degenerate quadratic form, 332 relative Reynolds operator, 398 strong rigidity theorem, 32 rigid analytic space, 451 symmetric algebra, 394 rigid braid cycle, 247 symmetry group, 31, 63 rigid braid cycle theorem, 247 symplectic group, 108 rigid braid orbit theorem, 216 rigid class vector, 29 Tate algebra, 449 H V rigid S -orbit, 48, 64 thick normal subgroup, 185 V rigid Hs -orbit, 212 transference, 347 rigidity defect, 263 translation theorem, 58 ring of holomorphic functions, 450 transvection, 100 ring of invariants, 394 trivial cover, 461 robust generating systems, 407 trivial F-module, 385 root, 93 twisted braid orbit theorem, 239 twisted rigidity theorem, 50 s-th V -symmetrized braid orbit genus, 235 twisted structure sheaf, 455 Scholz embedding problem, 374 twisted upper bound theorem, 423 Scholz extension, 374 Scholz solution, 374 uniform function, 126 Schur multiplier, 226 unipotent character, 127 second embedding obstruction, 364 uniquely liftable, 318 semiabelian group, 299 unirational function field, 200 semirational class, 41 universally central embeddable Galois shape function, 226 extension, 328 socle of an `-Galois extension, 375 unramified, 187 solution field of an embedding problem, 288 unramified rational place, 223 solution field of an F-module, 385 upper bound theorem, 391 solution of an embedding problem, 288 solution space of an F-module, 385 V -configuration, 48, 232 specialization theorem, 224 V -rigid class vector, 48 sphere relations, 182 V -symmetric, 31 spinor norm, 110 V -symmetrized braid orbit, 211 split embedding problem, 288 V -symmetrized irrationality degree, 31 splitting theorem, 13, 196 stability condition, 260 wreath extension, 347