Appendix: Example Polynomials

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 ofsmall 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 Galoisgroups over Q.t/. (There are 301 inequivalent transitive permutation groups ofdegree 12.) Most of these results are new. In the second table, we collect the knownexplicit regular Galois realizations of primitive non-solvable permutation groups ofdegree at most 31 over Q.t/ from the literature. For both sets of tables the resultswere mainly obtained by the rigidity method described in Chapter I and descentarguments.

Finally, we give example polynomials generating number fields over Q withgiven Galois group of permutation degree at most 14. For degree less than 12, thesewere either found by a random search, and then the Galois group was verified bythe Galois group recognition programs in several computer algebra systems, or theywere obtained by specializing the parametric realizations from the first set of tables.(Such specializations tend to have larger field discriminant.) The polynomials ofdegree 12 to 14 are taken from Kluners and Malle (2000, 2002). The polynomialslisted in this table were chosen so that their coefficient sum is small.

1 Regular Realizations for Transitive Groups of Degree Lessthan 12

Here we give polynomials generating regular field extensions of Q.t/ with Galoisgroups most of the transitive permutation groups of degree less than 12. The genericformulas for polynomials with symmetric or alternating group of arbitrary degreeare given separately. In all other cases the groups are numbered according to thelist in Butler and McKay (1983), so that a polynomial fn;i has Galois group thetransitive permutation group of degree n denoted by Ti in loc. cit.

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

491

https://doi.org/10.1007/978-3-662-55420-3

492 Appendix: Example Polynomials

Table 1.1 Symmetric and alternating groups

Sn xn � t.nx �n C 1/

An

(fSn

.x;1� .�1/n.n�1/=2nt2/ for n � 1 .mod 2/

fSn.x; 1=.1C .�1/n.n�1/=2.n �1/t2// for n � 0 .mod 2/

Table 1.2 Degree 4

f4;3 x4 �2x2 C t

f4;2 x4 C tx2 C 1

f4;1 x4 C tx3 �6x2 � tx C 1

Table 1.3 Degree 5

f5;3 x5 C 10x3 C 5tx2 �15x C t2 � t C 16

f5;2 x2.x C 1/2.x C 2/ � .x �2/2.x �1/t

f5;1 x.x2 �25/2 C .x4 �20x3 �10x2 C 300x �95/t2 �4.x �3/2t4

Table 1.4 Degree 6

f6;14 x6 �2x5 C .5x2 �6x C 2/t

f6;13 x6 � .3x �2/2 t

f6;12 f6;14.x;1�5t2/

f6;11 x6 � .3x2 C 1/t=4

f6;10 f6;13.x;1=.t2 C 1//

f6;9 f6;13.x;1� t2/

f6;8 f6;11.x;1=.1�3t2//

f6;7 f6;11.x; t2/

f6;6 f6;11.x;3t2 C 1/

f6;5 f6;13.x;�12t2.3t2 C 1//

f6;4 f6;11.x;.t2 C 3/2=.t2 �3/2/

f6;3 x2.x2 C 3/2 C 4t

f6;2 f6;3.x;3t2 C 1/

f6;1 x6 C .3x2 C 4/2.3t2 C 1/

Table 1.5 Degree 7

f7;5 .x4 �3x3 �x C 4/.x3 �x C 1/ �x2.x �1/t

f7;4 x7 C 28x6 C 63x5 C 1890x3 C 3402x2 �5103x C 33534C x.x6 �63x4

�3402x �5103/t C 13122t2

f7;3 see Smith (1993)

f7;2 f7;4.x;.t3 �27t2 �9t C 27/=.3.t3 C t2 �9t �1///

1 Regular Realizations for Degree Less than 12 493

f7;1 x7 �21�7.t/x5 �7�7.t/.10t3 C 5t2 �5t �3/x4 �7.15t6 C 15t5 �20t4 �27t3

�13t2 �6t �13/�7.t/x3 �7.12t9 C 18t8 �30t7 �63t6 �35t5 �14t4 �35t3

C2t2 C 31t C 16/�7.t/x2�7.t �1/�7.t/.5t11 C 15t10 �5t9 �62t8 �93t7

�91t6 �126t5 �166t4 �113t3 �30t2 �8t �12/x � .6t15 C 15t14 �35t13

�126t12 �63t11 C 70t10 �91t9 �271t8 C 131t7 C 427t6 C 126t5 �84t4

C175t3 C 189t2 �29t �97/�7.t/

Table 1.6 Degree 8

f8;48 x4.x �2/2.x2 C x C 2/ � .x �1/2.x2 C x C 1/t

f8;47 x8 � .4x �3/2 t

f8;46 f8;47.x; t2 C 1/

f8;45 f8;47.x;1=.1� t2//

f8;44 x8 C .4x2 C 3/t

f8;43 x6.x2 �x C 7/ �108.x �1/t

f8;42 f8;47.x;�1=.12t2.3t2 �1///

f8;41 .x2 �2/4 �26.2x �3/2t=33

f8;40 x4.x4 �8x2 C 18/ �27t

f8;39 f8;44.x;3t2/

f8;38 f8;44.x;1=.3t2 C 1//

f8;37 x8 C 6x7 C 3.7x2 C 6x C 36/.7t2 C 144/

f8;36 .f9;32.x;0/ �f9;32.t;0//=.x � t/

f8;35 f8;44.x;�27t2.t �1/=4/

f8;34 f8;41.x;1� t2/

f8;33 f8;41.x;1=.3t2 C 1//

f8;32 f8;40.x;�3t2/

f8;31 f8;44.x;27.t2 �1/2=.t2 C 3/3/

f8;30 x4.x4 C 4x2 C 6/ � .4x2 C 1/.3t2 C 2/2.3t2 �1/=4

f8;29 f8;44.x;27t2.t2 �1/2=4/

f8;28 f8;44.x;27t4.t2 C 1/=4/

f8;27 f8;44.x;27.t2 C 4/=.4.t2 C 3/3//

f8;26 f8;40.x;27t2.t �1/=..3t C 1/.3t �2/2//

f8;25 see Smith (1993)

f8;24 .x2 C x C 1/4 � .2x C 1/2 t

f8;23 x2.x2 C 396/2.x2 C 11/ � .x2 C 4/2.x2 C 256/t

f8;22 f8;41.x;27t2=.4.t2 �1/3//

f8;21 x8 C 2.t2 �1/x6 C .3t4 � t2/x4 C 2.t6 C t4/x2 C t8 C t6

f8;20 f8;41.x;27t4.t2 C 1/=4/


f8;19 f8;41.x;27.t2 C 4/=.4.t2 C 3/3//

f8;18 f8;41.x;27.t2 �1/2=.t2 C 3/3/

f8;17 .x4 C 4x3 �6x2 �4x C 1/2 �16x2.x2 �1/2 t

f8;16 f8;40.x;�27.t2 C 2/2=.t4.4t2 C 9///

f8;15 x8 C 8x6 C 4.4t �11/x4 C 8.t �3/.t �2/x2 C t.t �3/2

f8;14 f8;24.x;1�3t2/

f8;13 f8;24.x;1=.3t2 C 1//

f8;12 f8;23.x;�t2/

f8;11 f8;15.x; t2/

f8;10 f8;41.x;2233t4.t2 C 9/.t2 C 1/=.t2 C 3/6/

f8;9 f8;24.x;�27t2.t �1/=4/

f8;8 f8;15.x;.8t2 C 3/=.2t2 C 1//

f8;7 f8;15.x;4=.t2 C 1//

f8;6 f8;15.x;2t2 C 3/

f8;5 x8 �4.t2 C 2/.t2 C 1/x6 C 2.3t2 C 1/.t2 C 1/.t2 C 2/2x4

�4.t2 C 2/2.t2 C 1/3x2t2 C .t2 C 2/2.t2 C 1/4t4

f8;4 .x4 �6x2 C 1/2 C 16x2.x2 �1/2 t

f8;3 f8;24.x;27.t2 �1/2=.t2 C 3/3/

f8;2 f8;24.x;27.t2 C 4/=.4.t2 C 3/3//

f8;1 x8 �4.t4 C 1/x6 C 2.4t2 C 1/.t4 C 1/x4 �4.t2 C 1/.t4 C 1/t2x2 C .t4 C 1/t4

Table 1.7 Degree 9

f9;32 x9 �3x8 C 4x7 �28x6 C 126x5 �266x4 C 308x3 � t x2 C .3t �539/x

�4t C 805

f9;31 x4.x C 1/3.x C 3/2 �4=27.3x C 1/3 t

f9;30 x4.x �3/2.x3 �3x2 �12/ C 28 t

f9;29 f9;31.x; t2/

f9;28 f9;31.x;1=.3t2 C 1//

f9;27 f9;32.x;.t3 �6t2 C 3t C 1/=.t3 �3t C 1//

f9;26 .x3 �19x2 C 97x �27/.x2 �4x �7/3 C 16=27x2.x �7/t

f9;25 f9;30.x;1=.3t2 C 1//

f9;24 x6.x3 C 9x C 6/ �4.3x C 2/3 t

f9;23 f9;26.x;�.43923t2 C 18225/=.3t2 C 1//

f9;22 f9;24.x;1=.3t2 C 1//

f9;21 x4.x C 1/2.x C 2/2.x C 3/ �1=35x2.9x2 C 20x C 12/t C 1=39 t2

f9;20 f9;24.x;3t2 C 1/

f9;19 .f10;35.x;0/.t �1/2 �f10;35.t;0/.x �1/2/=.x � t/

f9;18 x6.x C 1/2.x �2/ C 4=27.3x C 2/3 t

f9;17 f9;21.x;25=.3t2 C 1//


f9;16 .x2 C x �2/4.x �4/ C 2433x3 t

f9;15

f9;14

f9;13 f9;18.x;1=.3t2 C 1//

f9;12 f9;18.x;�t2/

f9;11 f9;18.x;3t2 C 1/

f9;10 .x6 C 3x4 C 10x3 C 6x C 25/.x3 C 3x2 C 2/ C .x �1/.x C 2/

�.x3 �3x2 �6x �1/ � .x4 �7x3 C 6x2 �13x �14/t=9C 18t2

f9;9 f9;16.x; t2 C 1/

f9;8 f9;16.x; t2/

f9;7 .x3 C 27x2 �9x �27/.x2 C 3/3 �27=4.x2 �1/2x.x2 �9/.3t2 C 49/

f9;6 f9;21.x;96.t2 �9/2t2=..t4 �2t2 C 49/.3t2 C 1///

f9;5 f9;16.x;.t2 C 1/2=.t2 �1/2/

f9;4 f9;16.x;1=.3t2 C 1/2/

f9;3 f9;10.x;�.t3 C 6t2 C 3t �1/=.t3 �3t �1//

f9;2 x9 �6.t2 C 3/x7 �6x6t C 9.t4 C 9t2 C 9/x5 C 24t.t2 C 3/x4

�.4t6 C 69t4 C 213t2 C 81/x3 �216t3x2 C 12t2.3t4 �11t2 C 21/x �8t3

f9;1 x9 �27�9.t/x7 �54t.t2 �1/�9.t/x6 C 243�9.t/.2t4 C t3 � t2 C 1/x5

C243t.t2 �1/�9.t/ � .4t4C2t3 � t2CtC3/x4 �81.33t8C33t7 �26t6 �6t5

C69t4C16t3 �36t2 �3tC10/ ��9.t/x3 �2187t.t2 �1/.2t8C2t7 � t6 � t5

C4t4C3t3 � t2C1/�9.t/x2C729.2t3C1/.3t9C9t8C2t7 �14t6C17t4

Ct3 �9t2 � tC1/�9.t/xC243�9.t/.36t13C18t12 �60t11C30t10C64t9

�81t8 �9t7C87t6 �36t5 �54t4C21t3C15t2 �3t �1/

Table 1.8 Degree 10

f10;43 x10 � .5x �4/2 t

f10;42 f10;43.x;1=.t2 C 1//

f10;41 f10;43.x;1� t2/

f10;40 f10;43.x;�20t2.5t2 �1//

f10;39 x10 �55.x2 C 4/t

f10;38 f10;39.x;1=.t2 C 1//

f10;37 .x2 �4/5 �55x2 t

f10;36 f10;39.x;1�5t2/

f10;35 x10 �2x9 C 9x8 �729.x �1/2 t

f10;34 f10;39.x;4.t2 C t �1/2=.5.t2 C 1/2//

f10;33 .x �2/2.x2 C x �1/4 � .380x6 �784x5 C 300x4 C 360x3 �315x2

C60x C 4/t C 4.5x �4/2 t2

f10;32 f10;35.x; t2/

f10;31 f10;35.x;1=.2t2 C 1//


f10;30 f10;35.x;1�2t2/

f10;29 x10 C 10x6 �5tx4 �15x2 � t2 C t �16

f10;28 .x �2/.x �1/.x4 C x3 C 6x2 �4x C 1/.x2 C x �1/2 C .4x5 �20x2

C15x �2/ � .10x3 �10x2 C 1/t C .5x �4/.8x5 �40x2 C 35x �8/t2

f10;27 f10;33.x;�t=.4.t2 �1///

f10;26 f10;35.x;.t2 �2/2=.t2 C 2/2/

f10;25 f10;39.x;28t2=..t4 C 6t2 C 25/.t2 C 1/4//

f10;24 f10;29.x; t.t �8/=.t2 �1//

f10;23 x2.x4 �25/2 C .x8 �20x6 �10x4 C 300x2 �95/t �4.x2 �3/2t2

f10;22 x10 C 55.x2 C 256/4t

f10;21 .x2 C 9x C 24/2.x2 C 4x C 64/2.x2 �6x C 144/ �55x4.x C 8/2 t=4

f10;20

f10;19 .x2 C 1/4.x2 C 16/ �5.x7 C 11x5 �15x4 �5x3 C 38x2 �15x �7/t

C.x5 C 10x3 �15x �15x2 C 28/t2

f10;18 f10;28.x; t2/

f10;17

f10;16 f10;23.x; t2 �95=36/

f10;15 f10;23.x;95=.t2 �36//

f10;14 f10;23.x; t2/ D f5;1.x2; t/

f10;13 .x2 �5/5 �55.x2 C 5x C 6/4 t=4

f10;12 f10;22.x;1=.1� t2//

f10;11 f10;22.x;1�5t2/

f10;10 f10;21.x;4.3t2 C 32/=.4t2 C 1//

f10;9 f10;21.x;�4.5t2 �32//

f10;8 f10;23.x;.7t2 �24t C 7/2=.36.t2 �1/2//

f10;7 f10;13.x;1�5t2/

f10;6 f10;19.x;�4=.5t4 C 5t2 C 1//

f10;5 f10;22.x;�4t5.t �10/=.55.t2 C 2t C 5///

f10;4 f10;13.x;�4t5.t �10/=.55.t2 C 2t C 5///

f10;3 f10;22.x;�.11t2 C 4t �11/.t2 C 4t �1/5=.55.t2 C 1/2.t2 �1/4//

f10;2 x10 �2.t2 �125/x8 C .t2 �125/.t2 �4t �65/x6 �4.t2 �125/2.t �10/x4

C4.t2 �14t C 25/.t2 �125/2x2 �64.2t �25/.t2 �125/2

f10;1 x10 �20�10.t/x8 C 10.7t4 �7t3 C 17t2 �17t C 12/�10.t/x6

�25.4t8 �8t7C12t6 �16t5C25t4 �46t3C67t2 �38tC9/�10.t/x4

C5�10.t/.13t12�39t11C18t10C50t9 �125t8C376t7

�453t6 �214t5C1050t4 �1125t3C613t2 �164tC18/x2

��10.t/.�1�3tC32t2 �36t3 �10t4C34t5�13t6 �8t7C4t8/2


Table 1.9 Degree 11

f11;6 .fM12.x;0/.2t �1/2 �fM12

.t;0/.2x �1/2/=.x � t/

f11;5 x11 �3x10 C 7x9 �25x8 C 46x7 �36x6 C 60x4 �121x3 C 140x2 �95x C 27

Cx2.x �1/3 t

f11;4

f11;3 x11 �11.t2 C 11/x9 C 44.t2 C 11/2x7 �77.t2 C 11/3x5 C 55.t2 C 11/4x3

�11.t2 C 11/5x �2t.t2 C 11/5

f11;2

f11;1 x11 �55�11.t/x9 �11.30t5 C 15t4 �30t3 �25t2 �4t C 3/�11.t/x8

�11.90t10C90t9�240t8�350t7�229t6�97t5C35t4C13t3�42t2

�42t�75/�11.t/x7�11.168t15C252t14�840t13�1750t12�1218t11

�242t10C880t9C1265t8C880t7C836t6C572t5C437t4C430t3

C224t2C78t�36/�11.t/x6�11.210t20C420t19�1680t18�4550t17

�2723t16C2118t15C7971t14C11976t13C9282t12C6555t11C6523t10

C5466t9C6103t8C4089t7�422t6�2128t5�1887t4�722t3C355t2

C508tC452/�11.t/x5�11.180t25C450t24�2100t23�7000t22

�3080t21C10615t20C27060t19C40865t18C32857t17C10109t16

�2398t15�10128t14�6994t13�882t12�14413t11�33099t10�42438t9

�36861t8�18117t7�550t6C6589t5C2640t4�1063t3�958t2�648t

C117/�11.t/x4�11.105t30C315t29�1680t28�6650t27�1659t26

C20003t25C44905t24C64445t23C44116t22�34353t21�102124t20

�135499t19�138713t18�92626t17�79067t16�119189t15�147399t14

�166843t13�136359t12�38237t11C44396t10C74899t9C52267t8

C2031t7�22096t6�12051t5C3115t4C7001t3C1543t2�1896t

�1160/�11.t/x3�11.40t35C140t34�840t33�3850t32�154t31

C19008t30C39600t29C49203t28C14520t27�120945t26�280357t25

�348952t24�314514t23�145540t22C29359t21C33825t20�27126t19

�75933t18�85096t17C57717t16C276738t15C420112t14C438965t13

C296100t12C50632t11�97383t10�69608t9C16104t8C68277t7

C54527t6�3025t5�25355t4�7986t3C3117t2C2302t�84/�11.t/x2

�11.9t40C36t39�240t38�1250t37C227t36C9128t35C17905t34

C16150t33�12716t32�122980t31�290048t30�377822t29

�311551t28C1083t27C489620t26C744371t25C662921t24C433805t23

C172463t22C209836t21C561407t20C810964t19C909892t18

C777874t17C289801t16�197823t15�430310t14�356065t13�11405t12

C252280t11C230131t10C47388t9�93665t8�90187t7�24467t6

C19479t5C19576t4�4165t3�5861t2C1587tC999/ ��11.t/x

�.10t45C45t44�330t43�1925t42C792t41C19448t40C36036t39

C13761t38�83787t37�449020t36�1138951t35�1569333t34

�1270152t33C131912t32C3449677t31C7101292t30C8022157t29


C6359584t28 C 2562879t27 �1238875t26 C 266530t25 C 4792381t24

C7758954t23 C 9292575t22 C 6341588t21 C 107481t20 �3610200t19

�4999456t18 �3552868t17 C 1494614t16 C 4899972t15 C 3760834t14

C620191t13 �2831935t12 �4464948t11 �2695792t10 �670956t9

�36608t8 C 325281t7 C 187935t6 C 13585t5 C 170786t4 C 81906t3

�42372t2 �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 groupsof degree d with 12 � d � 31. Simple groups in this range for which no polynomialover Q.t/ is known to date are L2.16/, M23, L2.25/ and L2.27/. The polynomialswere taken from Hafner (1992), Konig (2015), Malle (1987, 1988a, 1993a), Malleand Matzat (1985), Matzat (1987), Matzat and Zeh-Marschke (1986) and Muller(2012). We also present surprisingly small polynomials of degree 100 with groupsAut.HS/ and HS taken from Barth and Wenz (2016). A polynomial of degree 266for the Janko group J2 has been obtained in Barth and Wenz (2017). In addition wegive the polynomial with Galois group Z16 from Dentzer (1995a).

Table 2.1 Primitive groups

M12 x12 C 44x11 C 754x10 C 6060x9 C 18870x8 �28356x7 �272184x6

�57864x5 C 1574445x4 �92960x3 �1214416x2 C 1216456x

�304119�492075.2x �1/2 t

PGL2.11/ .x3 �66x �308/4 �9t.11x5 �44x4 �1573x3 C 1892x2 C 57358x

C103763/ �3t2.x �11/

L2.11/ fPGL2.11/.x;2835=.11t2 C 1//

L3.3/ .x6 �6x4 C 64x3 �36x2 C 216/.x4 C 8x3 �108x2 C 432x �540/

�.x3 �18x2 C 54x �108/ � .3x4 �28x3 C 108x2 �216x C 108/2

�.x4 C 8x3 C 108/t

PGL2.13/ .x3 �x2 C 35x �27/4.x2 C 36/ �4.x2 C 39/6.7x2 �2x C 247/t=27

L2.13/ fPGL2.13/.x;1=.39t2 C 1//

PGL2.17/ .x3 �7x2 C 5x �2/6 � .x17 �17x15 C 34x14 C 85x13 �408x12 C 289x11

C1190x10 �2907x9 C 1462x8 C 3281x7 �5780x6 C 3196x5 C 238x4

�646x3 �68x2 C 120x �16/t C t2

L2.17/ fPGL2.17/.x;223317=.t2 �17//

PGL2.19/ .x5 C 26x4 C 69x3 C 108x2 C 68x C 16/4 � .x19 �38x17 �38x16

C513x15 C 1064x14 �2299x13 �9538x12 �5358x11 C 24358x10

C55081x9 C 35416x8 �40204x7 �105374x6 �98496x5 �41040x4

C3648x3 C 11552x2 C 4352x C 512/t C t2

L2.19/ fPGL2.19/.x;2819=.t2 C 19//

P�L3.4/ .x3 �9x2 �21x C 5/5.x C 1/5x � t.20x5 C 89x4 C 68x3 �50x2

C16x C 1/3 � .x5 C 57x4 C 330x3 C 914x2 C 1509x C 1125/

L3.4/:3 see Konig (2015)

L3.4/:22 .fAut.M22/.x;0/.t2 � t C 3/11 � .fAut.M22/.t;0/.x2 �x C 3/11/=.t �x/

L3.4/ see Konig (2015)


Aut.M22/ .5x4 C 34x3 �119x2 C 212x �164/4.19x3 �12x2 C 28x C 32/2

�222.x2 �x C 3/11 t

M22 fAut.M22/.x;1=.11t2 C 1//

M24 4.48x10 �192x9 �256x8 C 1104x7 C 520x6 �1276x5 �64x4 �776x3

�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

PGL2.23/ .x8 C 3x7 C 37x6 �24x5 C 121x4 C 333x3 C 429x2 C 216x C 36/3

�.2x24Cx23 �322x22C1219x21C1863x20C4094x19C99084x18

C197501x17C877910x16C1337726x15C3132117x14C8697795x13

C15394935x12C16590866x11C4182642x10C6982731x9

C36934642x8C43085601x7C13510591x6 �9423054x5

�10152936x4 �4024080x3 �824688x2 �85536x �3456/t

C.x24 �7x23C69x22 �460x21 �1564x20 �3289x19C11017x18

C19159x17 �20792x16 �269307x15 �650440x14 �547124x13

C609937x12C2106294x11C2682306x10C1410682x9 �856612x8

�1557215x7 �609132x6C135079x5C225814x4C113436x3

C33764x2C5904xC496/t2 � .x23C23x20C23x19C23x18

C161x17C368x16C529x15C575x14C1610x13C3036x12

C2668x11C2300x10C3542x9C5428x8C2599x7 �1748x6

�1265x5C345x4 �598x2 �252x �16/t3Ct4

L2.23/ fPGL2.23/.x;.23�33t2/=.t2 C 23//

U4.2/:2 .x3 C 6x2 �8/9 �24312x6.x2 C 5x C 4/4.x �2/t

U4.2/ .x3 C 6x2 �8/9 �24312x6.x2 C 5x C 4/4.x �2/.3t2 C 1/

S6.2/ .x4 �10x2 �8x C 1/7 �x3.x2 C 3x C 1/5 t

U3.3/:2 .x6 �6x5 �435x4 �308x3 C 15x2 C 66x C 19/4.x4 C 20x3 C 114x2

C68x C 13/ �2239.x2 C 4x C 1/12.2x C 1/t

U3.3/ fU3.3/:2.x;1=.t2 C 1//

PGL2.29/ .x5 �7x4 C 8x3 �17x2 C 9x �6/6 � t.x29 C 29x26 �29x25 C 29x24

C290x23 �638x22 C 899x21 C 464x20 �4118x19 C 8323x18

�9686x17 �899x16 C 20532x15 �46197x14 C 55477x13 �36801x12

�8584x11 C 66874x10 �100601x9 C 105560x8�73602x7 C 34017x6

�2349x5 �11745x4 C 10962x3 �6264x2 C 1944x �432/ C t2

L2.29/ fPGL2.29/.x;223329=.t2 �29//

PSL5.2/ .x5 �95x4 �110x3 �150x2 �75x �3/3.x5 C 4x4 �38x3 C 56x2

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

HS fAut.HS/.x;.5t2 C 1/=28/

2 Regular Realizations for Nonsolvable Primitive Groups 501

Table 2.2 The cyclic group Z16

Z16 x16 �24�16.t/x14 C 24.16t6 �14t4 C 6t2 C 5/�16.t/x12

�26.24t12 �28t10C6t8C36t6 �31t4C13t2C2/�16.t/x10

C25.128t18 �120t16 �144t14C560t12 �488t10C144t8C164t6

�136t4C56t2C1/�16.t/x8

�28.16t22C16t20 �120t18C208t16 �108t14 �64t12C164t10

�128t8C73t6 �20t4 C3t2C2/t2�16.t/x6

C28.64t24 �192t22C208t20C80t18 �432t16

C520t14 �316t12C112t10C18t8 �66t6C67t4 �26t2C5/t4�16.t/x4

�210.32t22 �112t20C160t18 �72t16 �84t14C144t12 �86t10C28t8

�17t6C17t4 �7t2C1/t6�16.t/x2

C28.8t10 �16t8C12t6 �4t2C1/2t8�16.t/


3 Realizations over Q for Transitive Groups of Degree up to 14

This last set of tables contains polynomials generating field extensions of Q withtransitive Galois group of degree less than fifteen. The polynomials are mainly takenfrom the database Kluners andMalle (2002), which contains polynomials for all buttwo transitive groups up to degree 23; see also Kluners and Malle (2000).

Table 3.1 Degree 2

T1 2 x2 C x C 1

Table 3.2 Degree 3

T2 S3 x3 �x �1

T1 3 x3 �x2 �2x C 1

Table 3.3 Degree 4

T5 S4 x4 �x C 1

T4 A4 x4 �2x3 C 2x2 C 2

T3 D4 x4 �x3 �x2 C x C 1

T2 V4 x4 �x2 C 1

T1 4 x4 C x3 C x2 C x C 1

Table 3.4 Degree 5

T5 S5 x5 �x3 �x2 C x C 1

T4 A5 x5 C x4 �2x2 �2x �2

T3 F20 x5 C x4 C 2x3 C 4x2 C x C 1

T2 D5 x5 �x3 �2x2 �2x �1

T1 5 x5 C x4 �4x3 �3x2 C 3x C 1

Table 3.5 Degree 6

T16 S6 x6 �x4 �x3 C x C 1

T15 A6 x6 �x3 �3x2 �1

T14 PGL2.5/ x6 �2x5 C 4x C 2

T13 32:D4 x6 C x5 �x2 �x C 1

T12 L2.5/ x6 �2x5 �5x2 �2x �1

T11 2� S4 x6 �x4 C 1

T10 32:4 x6 C x5 C x4 C x3 �4x2 C 5

T9 32:22 x6 �x3 C 2

T8 S4=4 x6 �x4 C 2x2 C 2

T7 S4=V4 x6 �x2 �1

T6 2� A4 x6 �3x2 C 1

T5 3� S3 x6 �3x3 C 3

T4 A4 x6 C x4 �2x2 �1

T3 D6 x6 �x3 �1

T2 S3 x6 C 3

T1 6 x6 �x3 C 1

3 Realizations over Q for Transitive Groups 503

Table 3.6 Degree 7

T7 S7 x7 C x3 �x2 C 1

T6 A7 x7 �2x6 C 2x C 2

T5 L3.2/ x7 �7x C 3

T4 F42 x7 �2

T3 F21 x7 �8x5 �2x4 C 16x3 C 6x2 �6x �2

T2 D7 x7 C 7x3 �7x2 C 7x C 1

T1 7 x7 �x6 �12x5 C 7x4 C 28x3 �14x2 �9x �1

Table 3.7 Degree 8

T50 S8 x8 C x4 C x C 1

T49 A8 x8 �8x3 C 10

T48 23:L3.2/ x8 �2x7 C 8x �2

T47 S4 o 2 x8 �5x �5

T46 x8 �8x3 �8x2 C 1

T45 x8 �3x4 �2x2 �4x �1

T44 2 o S4 x8 �x2 �1

T43 PGL2.7/ x8 �x7 C 7x6 �4x C 4

T42 A4 o 2 x8 �2x7 C 6x4 C 4

T41 x8 C 4x7 �2x4 �4x2 C 2

T40 x8 C 4x6 �9

T39 23:S4 x8 C x2 C 1

T38 2 o A4 x8 C 2x6 C 2x4 C 2

T37 L2.7/ x8 �4x7 C 7x6 �7x5 C 7x4 �7x3 C 7x2 C 5x C 1

T36 23:7:3 x8 C x7 C x6 �3x5 C 5x4 C 5x3 �7x C 9

T35 2 o 2 o 2 x8 C 2x6 C 2

T34 x8 �x7 C 2x6 �x5 �2x4 C 4x3 �6x C 4

T33 x8 �4x5 C 12x4 �8x2 C 12x C 9

T32 x8 C x6 C 3x2 C 4

T31 2 o 22 x8 C 4x6 �8x2 �1

T30 x8 �4x6 C 4x4 �2

T29 23:D4 x8 �x6 C x2 C 1

T28 x8 C 4x6 C 2

T27 2 o 4 x8 �8x4 C 8x2 �2

T26 x8 C x4 C 2

T25 23:7 x8 �4x7 C 8x6 �6x5 C 2x4 C 6x3 �3x2 C x C 3

T24 S4 � 2 x8 �4x2 C 4

T23 GL2.3/ x8 �6x4 �x2 �3

T22 x8 �x4 C 4

T21 x8 �2x6 C x4 C 5

T20 x8 �3x6 �x4 C 3x2 C 1

T19 x8 C 4x4 �4x2 C 1

T18 22 o 2 x8 �x6 C 2x2 C 1

T17 4 o 2 x8 �2x4 C 2

T16 x8 C 4x4 C 2

T15 x8 C 3

T14 S4 x8 C 4x6 C 4x2 C 4

T13 A4 � 2 x8 C 2x6 C 3x4 �3x2 C 1


T12 SL2.3/ x8 C 9x6 C 23x4 C 14x2 C 1

T11 x8 C 9

T10 x8 �2x6 C 4x4 �3x2 C 1

T9 D4 � 2 x8 C 4x4 C 1

T8 x8 �2

T7 x8 �15x4 C 10x2 C 5

T6 D8 x8 C 2

T5 Q4 x8 C 12x6 C 36x4 C 36x2 C 9

T4 D4 x8 C 3x4 C 1

T3 23 x8 �x4 C 1

T2 4� 2 x8 C 1

T1 8 x8 C x7 �7x6 �6x5 C 15x4 C 10x3 �10x2 �4x C 1

Table 3.8 Degree 9

T34 S9 x9 C x5 �x2 C 1

T33 A9 x9 �3x3 C x C 2

T32 �L2.8/ x9 C x7 C 2x5 C 4x3 �x2 C x C 1

T31 S3 o S3 x9 �x8 C 2x2 �x C 1

T30 x9 C 2x5 �4x4 C 4x3 �4x2 C x �1

T29 x9 �3x6 �5x5 C 5x2 �1

T28 S3 o 3 x9 �2x6 �4x3 C 3x C 1

T27 L2.8/ x9 C x7 �4x6 �12x4 �x3 �7x2 �x �1

T26 32:GL2.3/ x9 �x7 C 5x6 C x5 �2x4 C 4x3 C 3x2 �x �1

T25 x9 �3x6 C 9x5 �9x4 �27x3 C 9x C 1

T24 x9 �2x6 �2

T23 32:SL2.3/ x9 �3x8 C x6 C 15x5 �13x4 �3x3 C 4x �1

T22 x9 �3x6 C 3

T21 x9 �6x3 �6

T20 3 o S3 x9 �x6 �2x3 C 1

T19 x9 �3x8 C 18x5 C 18x4 �27x C 9

T18 x9 �x3 �1

T17 3 o 3 x9 C x8 �10x7 �14x6 C 20x5 C 36x4 �18x2 �8x �1

T16 32:D4 x9 �x8 �x5 �x4 C 3x3 C 2x2 �1

T15 32:8 x9 �4x8 C 8x7 �32x5 C 80x4 �104x3 C 80x2 �34x C 8

T14 32:Q4 x9 �12x5 C 132x �128

T13 x9 �3x3 �1

T12 x9 �2x8 C x5 �3x3 C 4x2 �12x C 8

T11 32:6 x9 �x6 C 5x3 C 1

T10 9:6 x9 �2

T9 32:4 x9 C 2x7 �3x6 C x5 �x4 C 64x3 �x �1

T8 S23 x9 C 3x3 �1

T7 32:3 x9 �3x8 �21x7 C 78x5 C 69x4 �21x3 �39x2 �12x �1

T6 9:3 x9 �14x7 C 63x5 �98x3 C 42x �7

T5 32:2 x9 �3x6 �3x3 �1

T4 S3 � 3 x9 �3x6 �6x3 �1

T3 D9 x9 �9x6 C 27x3 �3

T2 32 x9 �15x7 C 4x6 C 54x5 �12x4 �38x3 C 9x2 C 6x �1

T1 9 x9 �9x7 C 27x5 �30x3 C 9x �1


Table 3.9 Degree 10

T45 S10 x10 �x3 �1

T44 A10 x10 �2x9 C 3x5 �4

T43 S5 o 2 x10 C 3x6 �2x5 C 1

T42 x10 C 5x8 �5x7 C 5x6 �7x5 �5x4 �10x2 �4

T41 x10 �2x9 �x6 C x4 �4x2 C 2x �1

T40 A5 o 2 x10 �x9 �x4 �4x3 C 4x2 �x �1

T39 2 o S5 x10 �x2 C 1

T38 x10 �3x8 C 2x2 C 2

T37 24:S5 x10 �x2 �1

T36 2 o A5 x10 C x4 �2x2 C 3

T35 P�L2.9/ x10 �4x9 C 6x8 C 12x2 C 16x C 8

T34 24:A5 x10 C 4x4 C x2 �4

T33 .5:4/ o 2 x10 C 6x6 C 8x5 �35x2 C 24x C 16

T32 S6 x10 �2x9 C x8 �9x2 C 2x �1

T31 M10 x10 �2x9 C 9x8 �54x2 C 108x �54

T30 PGL2.9/ x10 �2x9 C 9x8 �7x2 C 14x �7

T29 2 o .5:4/ x10 C 10x6 C 5

T28 x10 �10x7 C 10x6 C 36x5 C 50x4 �10x3 �1

T27 x10 C 3x6 �2x5 C x2 C 2x C 1

T26 L2.9/ x10 �x9 C 3x8 �6x7 C 3x6 �3x5 �3x3 �6x2 �8x �1

T25 x10 C 10x6 �5

T24 24:5:4 x10 C 5x6 C 5x2 �1

T23 2 o .5:2/ x10 �5x4 �3

T22 S5 � 2 x10 C 4x2 C 4

T21 D5 o 2 x10 C x6 �2x5 �x4 C 3x2 �2x C 1

T20 52:Q4 x10 �10x8 C 35x6 �4x5 �50x4 C 20x3 C 25x2 �20x �17

T19 52:D4 x10 �10x8 C 35x6 �2x5 �50x4 C 10x3 C 25x2 �10x C 2

T18 52:8 x10 C 60x6 �208x5 C 850x2 �8000x �4672

T17 x10 C x5 C 2

T16 x10 �5x4 C 15

T15 24:5:2 x10 �5x4 �4x2 �1

T14 2 o 5 x10 C x8 �4x6 �3x4 C 3x2 C 1

T13 S5=D6 x10 �x9 �x8 C 3x6 �x5 �2x4 C 3x3 �x2 �x C 1

T12 S5=A4 x10 C 2x9 C 3x8 �x6 �2x5 �x4 C 3x2 C 2x C 1

T11 A5 � 2 x10 C x8 �4x2 C 4

T10 52:4 x10 �2x5 �4

T9 52:22 x10 �x9 �5x8 C 11x6 C 4x5 �10x4 C 25x2 C 5x �5

T8 24:5 x10 �4x8 C 2x6 C 5x4 �2x2 �1

T7 A5 x10 �x8 �4x7 �3x6 �2x5 C 8x3 �2x �1

T6 5 o 2 x10 �x9 C 3x7 �3x6 C x5 C 5x4 �x3 C 2x2 C 3x C 1

T5 2� 5:4 x10 C 2

T4 5:4 x10 �5

T3 D10 x10 �3x4 C 2x2 C 1

T2 D5 x10 C 5x8 C 15x6 C 20x4 C 25x2 C 15

T1 10 x10 C x9 C x8 C x7 C x6 C x5 C x4 C x3 C x2 C x C 1


Table 3.10 Degree 11

T8 S11 x11 C x6 C x4 C 1

T7 A11 x11 �6x8 C 4x5 �3x3 C 2

T6 M11 x11 �4x10 C 60x7 �108x6 C 72x5 �360x4 C 3636x �1944

T5 L2.11/ x11 �2x10 C x9 �5x8 C 13x7 �9x6 C x5 �8x4 C 9x3 �3x2 �2x C 1

T4 F110 x11 �3

T3 F55 x11 �33x9 C 396x7 �2079x5 C 4455x3 �2673x �243

T2 D11 x11 �x10 C 5x8 C 8x5 C 6x4 �x3 C x2 C 3x C 1

T1 11 x11 C x10 �10x9 �9x8 C 36x7 C 28x6 �56x5 �35x4 C 35x3

C15x2 �6x �1


S12 x12 �x C 1

A12 x12 C 3x8 C 3x4 C 4x3 C 4

T299 x12 C x2 �2x C 1

T298 x12 �72x2 �120x �50

T297 x12 �2x7 C 7x6 C x2 �2x C 1

T296 x12 �x7 �7x6 �5x4 �x2 C x C 1

M12 x12 �375x8 �3750x6 �75000x3 C 228750x2 �750000x C 1265625

T294 x12 C 4x9 �6x7 C 2

T293 x12 �x6 �x2 �1

T292 x12 �3x11 C 5x9 �3x8 C 3x7 C 2x6 �6x5 �3x4 C 1

T291 x12 �12x9 �9x8 �64x3 �144x2 �108x �27

T290 x12 C 3x10 �x9 C 2x6 �3x5 C 9x4 �3x3 C 3x C 1

T289 x12 �4x9 C 2x6 C 4x4 C 1

T288 x12 �4x11 C 4x10 �50x4 C 120x3 �112x2 C 48x �8

T287 x12 �3x8 C 2x6 C 3

T286 x12 �x6 �3x4 �1

T285 x12 �4x2 C 4

T284 x12 �12x9 �9x8 C 64x3 C 144x2 C 108x C 27

T283 x12 �8x9 C 24x6 C 144x5 C 96x3 C 144x2 C 48

T282 x12 �x11 C 3x10 �x9 C 6x8 C 6x6 �2x5 C 7x4 C 4x3 C 4x2 C x C 1

T281 x12 �x11 C x10 �2x8 C 3x7 �3x5 C 3x4 �2x3 C 3x2 �x C 1

T280 x12 �4x11 C 6x10 �2x9 �5x8 C 6x7 �4x5 C 2x4 C 2

T279 x12 �4x11 C 4x10 C 4x7 �6x6 �4x5 C 36x2 C 36x C 9

T278 x12 C 20x8 �80x6 C 50x4 �320x3 �912x2 C 1280x C 800

T277 x12 C 3x6 C 3x2 C 4

T276 x12 C 192x6 �288x5 C 108x4 C 256x3 �576x2 C 432x �108

T275 x12 C x10 �9x9 C 11x8 �11x7 C 17x6 �7x5 C 2x4 C x C 1

T274 x12 �x8 C 2x6 �4x3 C 1

T273 x12 �12x9 C 9x8 C 192x3 �432x2 C 324x �81

M11 x12 C 6x11 C 15x10 C 28x9 C 36x8 C 6x7 �75x6 �108x5 C 18x4 C 82x3

C3x2 �6x C 5

T271 x12 �135x8 �180x7 C 399x6 C 918x5 C 693x4 C 352x3 C 216x2 C 96x C 16

T270 x12 C x10 C 4x2 �1

T269 x12 �2x10 �6x8 C 14x6 C x4 �8x3 C 1

T268 x12 C 4x9 �3x8 �64x3 C 144x2 �108x C 27


T267 x12 C 12x10 �8x9 C 54x8 �48x7 C 132x6 �72x5 �33x4 �32x3 C 8

T266 x12 C x10 �4x9 �2x8 �3x7 C 4x5 C 2x3 �2x2 �x C 1

T265 x12 �8x10 C 7x9 C 8x8 �7x7 �15x6 C 21x5 C 8x4 �14x3 �8x2 C 7x C 1

T264 x12 �4x11 C 6x10 �3x9 �2x8 C 3x7 �2x5 C x4 C x3 �x2 C 1

T263 x12 �162x4 �432x3 �432x2 �192x �32

T262 x12 C 18x8 �24x7 C 8x6 �81x4 C 216x3 �216x2 C 96x �16

T261 x12 C 2x6 �4x5 C x4 C 1

T260 x12 �3x2 C 3

T259 x12 �12x10 C 54x8 �110x6 C 93x4 �4x3 �18x2 C 12x �8

T258 x12 �x3 �3

T257 x12 C 4x10 �4x2 C 4

T256 x12 C 4x10 �5x2 C 5

T255 x12 �2x8 �6x6 C 9x4 �1

T254 x12 C 6x10 �12x9 �54x7 C 24x6 C 180x4 C 156x3 C 216x2 C 72x C 18

T253 x12 �4x9 �3x8 �32x6 �48x5 �18x4 C 64x3 C 144x2 C 108x C 27

T252 x12 �12x9 C 27x8 C 12x6 �36x5 C 27x4 �16x3 C 36x2 C 9

T251 x12 C 48x6 �72x5 C 27x4 C 64x3 �144x2 C 108x �27

T250 x12 C 3x2 C 5

T249 x12 �12x10 C 54x8 �108x6 C 81x4 �8x3 C 24x C 8

T248 x12 C 324x6 �648x5 C 675x4 �744x3 C 648x2 �288x C 48

T247 x12 �8x9 C 24x6 C 162x4 �32x3 C 16

T246 x12 C 81x4 �216x3 C 216x2 �96x C 16

T245 x12 �12x10 �54x8 �72x7 C 96x6 C 9x4 C 200x3 C 108x2 �4

T244 x12 �3x11 �6x10 C 13x9 C 6x8 �15x7 C 5x6 �15x5 C 15x4 C 5x3 �5

T243 x12 �9x8 �12x7 �4x6 �81x4 �216x3 �216x2 �96x �16

T242 x12 �4x9 C 18x8 �4x6 �36x5 C 81x4 C 16x3 C 108x2 C 16

T241 x12 C x10 �3x8 �x6 C 6x4 �3

T240 x12 C 6x8 C 4x6 �4

T239 x12 �12x9 C 9x8 �32x6 C 48x5 �18x4 �64x3 C 144x2 �108x C 27

T238 x12 C 6x10 C 9x8 �8

T237 x12 C 10x2 C 5

T236 x12 C x4 C 2x2 C 1

T235 x12 C 3x8 �4x6 C 2

T234 x12 C x9 C 3x3 C 4

T233 x12 �4x3 �6

T232 x12 �13x8 �26x7 �11x6 C 6x5 C 25x4 C 78x3 C 114x2 C 76x C 19

T231 x12 �x11 C 2x9 �x8 �4x7 C 5x6 �x5 �x4 �x3 C 4x2 �3x C 1

T230 x12 C x10 �3x8 C 4x4 C 1

T229 x12 �18x10 �22x9 C 102x8 C 180x7 �96x6 �90x5 C 81x4 �30x3

�54x2 C 3

T228 x12 C 4x11 C 3x10 �2x9 C 11x8 C 30x7 C 14x6 �11x5 C 12x4 C 30x3

Cx2 �9x �1

T227 x12 C 2x10 C x8 �4x4 C 3

T226 x12 �3x8 �6x2 C 1

T225 x12 �3x10 C 2x6 C 2x4 �3

T224 x12 C 4x8 C 6x6 �6x2 C 2

T223 x12 �6x10 C 12x6 �9

T222 x12 �4x6 C 3x2 �1

T221 x12 �2x10 �x8 C 6x6 �x4 �4x2 �1


T220 x12 �4x9 �12x8 C 34x6 �12x5 C 45x4 C 42x2 C 10

T219 x12 C 2x6 C x2 C 1

T218 x12 �2x11 C 22x9 �88x7 C 176x5 �176x3 C 64x C 4

T217 x12 �4x9 C 2

T216 x12 �12x10 �8x9 C 162x4 C 432x3 C 432x2 C 192x C 32

T215 x12 �3x10 C x9 �81x8 C 54x7 �36x6 C 27x5 C 72x4 �107x3

C54x2 �12x C 1

T214 x12 �12x9 C 18x8 �56x6 C 138x4 �96x3 C 72x2 C 72

T213 x12 �x3 �1

T212 x12 �12x10 �18x8 �96x7 �132x6 �63x4 �64x3 C 72x2 �16

T211 x12 C 90x8 C 120x7 C 40x6 C 405x4 C 1080x3 C 1080x2 C 480x C 80

T210 x12 �4x9 C 8x6 �36x5 C 105x4 �120x3 C 90x2 �36x C 9

T209 x12 �8x9 C 18x8 �24x7 C 24x6 �33x4 �16x3 �48x �8

T208 x12 �3x10 C 3x6 C 3x4 C 3

T207 x12 �x11 C x9 �x7 �x6 C 2x5 �x4 �3x3 C 3x2 �2x C 1

T206 x12 �12x9 C 15x8 �12x5 C 18x4 �64x3 C 96x2 �36x C 9

T205 x12 �208x6 �312x5 �117x4 �832x3 �1872x2 �1404x �351

T204 x12 �6x9 C 18x8 C 48x6 C 108x4 �32x3 �72x C 24

T203 x12 �2x6 C x4 C 1

T202 x12 �4x6 C 9x4 C 4

T201 x12 C 3x10 �12x2 C 24

T200 x12 C 6x10 C 9x8 �12

T199 x12 �2x10 �4x8 �x6 C x4 C 4

T198 x12 �2x10 �x8 C 6x4 �4x2 C 2

T197 x12 C 4x6 �9x4 C 8

T196 x12 �2x8 �4x6 C 6x4 C 4x2 �1

T195 x12 C 4x10 C 2x8 �4x6 C 4

T194 x12 C 2x10 C 2x9 �x8 �2x7 C 4x6 �12x5 C 6x4 C 2x3 C 18x2 C 27

T193 x12 C 6x6 C 6x4 C 3

T192 x12 �6x10 C x8 C 36x6 �30x4 �28x2 C 18

T191 x12 C x10 C 2x8 �x6 C 2x4 �3x2 C 1

T190 x12 C 2x10 �13x8 C 36x6 C 15x4 �38x2 �19

T189 x12 �6x8 C 12x4 C 13x2 C 5

T188 x12 �2x10 C 5x6 C 5x2 �1

T187 x12 C 8x6 �9x2 C 1

T186 x12 �x10 �x2 �1

T185 x12 �x4 �2

T184 x12 C x8 C 9x6 C 9x4 C 7x2 C 1

T183 x12 �7x6 �10x4 �5x2 C 1

T182 x12 �8x9 C 6x8 C 20x6 �24x5 C 18x4 �16x3 C 24x2 C 8

T181 x12 �18x8 �36x6 �72x5 C 54x4 �144x3 �216x2 �72

T180 x12 �2x10 C 5x8 �8x6 C 6x4 �4x2 C 1

L2.11/ x12 C x11�8x10 �29x9 C 48x8 C 51x7�5x6 C 275x5 C 642x4 C 208x3

C308x2 C 41x C 2

T178 x12 �x9 C 4x3 �1

T177 x12 �4x9 C 4x3 C 2

T176 x12 C 4x6 �8x3 C 8

T175 x12 �2x11 C 4x10 �2x9 C 4x7 �3x6 C 2x5 C x2 �2x C 1

T174 x12 C 12x10 C 54x8 C 20x6 �447x4 �384x3 �792x2 �1152x �368


T173 x12 �36x8 �48x7 �32x6 C 162x4 �288x2 C 128

T172 x12 C 12x10 �6x9 �54x7 �157x6 C 210x4 C 174x3 C 234x2 C 252x C 118

T171 x12 �8x9 �36x8 �72x5 C 81x4 C 64x3 �144x2 C 64

T170 x12 �x9 C 2x6 C 4x3 C 3

T169 x12 �8x3 C 18

T168 x12 �10x6 �12x3 �2

T167 x12 �3x3 C 3

T166 x12 C 18x10 C 135x8 C 348x6 C 63x4 �512x3 �270x2 C 729

T165 x12 �16x9 C 12x8 C 256x3 �576x2 C 432x �108

T164 x12 C 4x9 C 6x7 C 8x6 �54x5 C 88x3 �57x2 �90x C 111

T163 x12 �x8 �2x6 C x4 �2x2 C 1

T162 x12 �2x8 �8x6 C 14x4 �16x2 C 4

T161 x12 C 3x10 C 18x2 C 9

T160 x12 C x10 C x8 C x6 �4x4 C 5

T159 x12 C 4x10 �4x8 �24x6 �x4 C 32x2 C 8

T158 x12 �x8 �2x6 C 2x2 C 1

T157 x12 �8x9 C 24x7 C 44x6 �51x4 C 48x3 �72x2 C 16

T156 x12 �2x9 C 2

T155 x12 �2x10 �3x8 C 2

T154 x12 �2x6 C 12x4 �6x2 C 7

T153 x12 C 2x10 C 8x2 C 8

T152 x12 �4x8 �2x6 C 4x4 �1

T151 x12 �3x8 �2

T150 x12 �x6 �3x4 C 2x2 C 2

T149 x12 �9x4 �6

T148 x12 C 3x10 C 3x8 C x6 �3

T147 x12 �3x8 �8

T146 x12 �2x10 �x8 �2x6 �2x4 �8x2 C 8

T145 x12 C 6x8 C 4x6 �18x4 �24x2 �8

T144 x12 C 6x10 C 4x8 �24x6 �21x4 C 22x2 C 4

T143 x12 �6x10 C 24x8 �56x6 C 93x4 �90x2 C 51

T142 x12 C 3x8 C 4x6 C 6x4 C 3

T141 x12 C 3x8 �3

T140 x12 �x4 �4

T139 x12 C 3x10 C 3x2 C 1

T138 x12 �x4 C 1

T137 x12 C x8 �2x6 �x4 �1

T136 x12 �x10 C 4x2 C 1

T135 x12 �18x8 �24x6 C 27x4 C 36x2 �6

T134 x12 �7x10 C 14x8 �21x4 C 7x2 C 7

T133 x12 �8x9 C 162x8 �372x7 C 20x6 C 432x5 �63x4 �212x3 �36x2

C24x C 56

T132 x12 �x10 �11x9 C 99x8 �45x7 �117x6 �27x5 C 90x4 C 36x3 C 9x C 18

T131 x12 �2x11 �x10 C 9x9 �7x8 �11x7 C 20x6 C x5 �19x4 C 8x3

C6x2 �5x C 1

T130 x12 �2x9 C x6 C 6x3 C 3

T129 x12 �6x10 �2x9 C 3x8 �30x7 C 8x6 C 90x5 C 36x4 �24x3 C 6x �1

T128 x12 �12x10 �22x9 C 57x8 �72x6 C 30x5 C 15x4 �30x3 C 6x C 1

T127 x12 �16x9 C 18x8 �72x6 C 36x5 �36x4 �76x3 �72x �62

T126 x12 C x8 C x6 �2x4 �x2 C 1


T125 x12 �2x8 �2x6 C x4 C 2x2 �1

T124 x12 C 4x10 C 10x6 C 5

T123 x12 �2x10 C 10x6 �8x2 C 1

T122 x12 �2x11 �3x10 �6x9 C 21x8 �32x7 C 37x6 �16x5 C 11x4 C 32x3

�x2 C 20x C 1

T121 x12 �x9 C 2x3 C 1

T120 x12 �2x9 �6x3 C 9

T119 x12 �8x6 �8x3 �2

T118 x12 C 8x6 �8x3 C 2

T117 x12 �2x9 C x6 C 5

T116 x12 �2x9 C 4x3 C 4

T115 x12 �2x8 C 3x4 �4

T114 x12 �x4 �1

T113 x12 �x4 C 4

T112 x12 �3x8 C 9x4 C 1

T111 x12 �6x8 C 68x6 C 105x4 C 36x2 C 12

T110 x12 C x8 �x6 �x4 �1

T109 x12 C x10 �4x2 C 1

T108 x12 �3x8 �4x6 C 6x4 C 4

T107 x12 C 6x10 C 3x8 �28x6 �21x4 C 30x2 C 5

T106 x12 C 3x10 �2x8 �9x6 C 5x2 C 1

T105 x12 �7x10 C 7x8 C 14x6 �16x4 �5x2 C 5

T104 x12 C 6x10 C 12x8 C 8x6 �3x4 �6x2 �1

T103 x12 C 3x10 �x6 C 3x2 C 1

T102 x12 �5x10 C 20x8 �70x6 C 145x4 �280x2 C 208

T101 x12 �3x10 �3x2 C 1

T100 x12 �x10 C x8 C 4x6 �x4 �x2 �1

T99 x12 �76x8 C 325x6 �380x4 C 125

T98 x12 �64x10 �231x8 C 740x6 �481x4 C 37

T97 x12 C x8 C 9x4 C 1

T96 x12 �3x4 �4

T95 x12 �x10 C 3x6 �2x4 �3x2 C 1

T94 x12 �57x8 �38x6 C 318x4 �204x2 C 17

T93 x12 C 10x10 C 28x8 C 6x6 �43x4 C 6x2 C 3

T92 x12 �9x4 �9

T91 x12 C 5x10 C 9x8 C 8x6 C 2x4 �12x2 C 16

T90 x12 C 2x10 �x6 C 2x2 C 1

T89 x12 �3x4 C 1

T88 x12 �6x8 �4x6 �3x4 �18x2 C 3

T87 x12 C 6x10 C 9x8 �4x6 �12x4 C 1

T86 x12 C 2x8 �2

T85 x12 �3x11 �3x10 C 15x9 �15x8 �33x7 C 29x6 C 15x5 �30x4 �128x3

�30x2 C 198x C 48

T84 x12 �6x10 C 4x9 C 21x8 �12x7 �52x6 �16x3 C 48x2 C 16

T83 x12 C 3x6 �x3 C 3

T82 x12 �12x10 C 54x8 �116x6 C 129x4 �72x2 �16

T81 x12 C x6 C 2


T80 x12 �90x8 C 160x6 �135x4 C 7200x2 �80

T79 x12 C 4x10 C 6x8 C 4x6 C 2

T78 x12 �x9 C x3 C 1

T77 x12 �2x6 C 5x2 C 1

T76 x12 C 2x8 C 5x4 C 6x2 C 1

T75 x12 C 7x8 C 7x4 C 8x2 C 1

T74 x12 �x10 C 2x8 C 4x6 �3x4 �3x2 C 1

T73 x12 �3x11 C 4x10 �x8 �6x7 C 20x6 �10x5 C 8x4 C 24x3 C 3x2 C 12x C 9

T72 x12 �6x10 �10x9 C 36x8 �116x6 C 720x5 C 696x4 �2440x3 �720x2

C1200x C 880

T71 x12 �4x9 C 4x6 C 3

T70 x12 C 9x6 �18x3 C 9

T69 x12 �3x10 �2x8 C 9x6 �5x2 C 1

T68 x12 C x10 C 6x8 C 3x6 C 6x4 C x2 C 1

T67 x12 �x8 �x6 �x4 C 1

T66 x12 C 6x10 C 12x8 C 8x6 �3

T65 x12 �3x4 C 4

T64 x12 C 3x8 �16

T63 x12 �6x10 C 104x6 C 93x4 C 18x2 C 4

T62 x12 �3x10 C 3x8 �x6 C 4x4 �4x2 C 1

T61 x12 �3x4 �1

T60 x12 �4x8 �9x4 C 4

T59 x12 �6x10 C 6x8 �4x6 �3x4 C 3

T58 x12 �12x8 �14x6 C 9x4 C 12x2 C 1

T57 x12 C 38x10 C 533x8 C 3474x6 C 10574x4 C 12740x2 C 4225

T56 x12 �2x10 C x6 �2x2 C 1

T55 x12 C 2x10 �97x8 �360x6 �345x4 �50x2 C 25

T54 x12 �6x8 C 9x4 C 2

T53 x12 C 2x8 �16x6 C 4x4 C 8

T52 x12 �3x4 �6

T51 x12 C 6x8 C 9x4 C 3

T50 x12 �3x4 C 6

T49 x12 C 3x8 �4x6 �3x4 �1

T48 x12 C 8x4 C 1

T47 x12 �6x10 C 20x9 �72x7 C 128x6 �96x5 C 45x4 �8x3 �18x2 C 12x �2

T46 x12 �4x11 C 6x10 C 4x9 �21x8 C 40x7 �28x6 �8x5 C 25x4 �28x3

C10x2 �4x �1

T45 x12 �3x9 �18x8 �24x6 �9x5 C 69x4 �x3 C 3x �1

T44 x12 �6x6 �10x3 �6

T43 x12 �6x9 C 10x6 C 4x3 C 2

T42 x12 �x6 C 7

T41 x12 �x9 �6x6 C x3 C 1

T40 x12 �7x10 C 24x8 �36x6 C 24x4 C 13x2 C 1

T39 x12 �4x6 C 2

T38 x12 C x6 �3

T37 x12 C x6 C 4

T36 x12 �2x9 �2x3 C 1

T35 x12 �x9 �x6 C x3 C 1


T34 x12 C 12x10 C 54x8 C 108x6 C 81x4 C 16

T33 x12 C 2x8 C 58x6 C 301x4 C 174x2 C 25

T32 x12 C 7x10 �x8 �23x6 �x4 C 7x2 C 1

T31 x12 C 6x10 �23x8 �210x6 �360x4 �50x2 C 25

T30 x12 �7x10 �14x8 C 115x6 �70x4 �175x2 C 125

T29 x12 �45x8 C 50x6 C 225x4 �375x2 C 125

T28 x12 C 2

T27 x12 C 12x10 C 68x8 C 220x6 C 392x4 C 360x2 C 148

T26 x12 �9x8 �8x6 �9x4 C 1

T25 x12 C 5x8 C 6x4 C 1

T24 x12 �2x8 �7x4 C 16

T23 x12 �4x4 C 4

T22 x12 �5x10 C 7x8 �6x7 �17x6 �6x5 C 7x4 �5x2 C 1

T21 x12 C 3x8 �4x6 C 3x4 C 1

T20 x12 �4x9 C 72x8 �84x7 C 236x6 �144x5 C 324x4 �192x3 C 72x2 C 8

T19 x12 C 24x10 C 196x8 C 600x6 C 452x4 C 112x2 C 8

T18 x12 C 2x6 C 4

T17 x12 C 4x8 C 4x6 C 5x4 C 12x2 C 2

T16 x12 �x6 C 4

T15 x12 C 3

T14 x12 �9x6 C 27

T13 x12 �3

T12 x12 C x6 �27

T11 x12 �8x6 C 8

T10 x12 C 9

T9 x12 C 3x8 C 4x6 C 3x4 C 1

T8 x12 �6x10 �8x9 C 9x8 C 12x7 �20x6 C 9x4 �24x3 �4

T7 x12 C 4x10 �x8 �x4 C 4x2 C 1

T6 x12 C 2x10 �6x8 C 2x6 �6x4 C 2x2 C 1

T5 x12 �80x10 C 1820x8 �13680x6 C 29860x4 �2720x2 C 32

T4 x12 C 6x8 C 26x6 �63x4 C 162x2 C 81

T3 x12 C 36

T2 x12 �x6 C 1

T1 x12 �x11 C x10 �x9 C x8 �x7 C x6 �x5 C x4 �x3 C x2 �x C 1


T9 S13 x13 �x C 1

T8 A13 x13 C 156x �144

T7 L3.3/ x13 C x12 C 40x10 C 13x9 �99x8 C 180x7 �468x6 �468x5 C 1644x4

�912C 24x C 24

T6 F156 x13 �2

T5 F78 x13 C 3x9 �10x8 �3x7 C 5x6 �20x5 �11x4 C 2x3 �10x2 �10x �3

T4 F52 x13 C 13x10 �26x8 C 13x7 C 52x6 �39x4 C 26x2 C 13x C 2

T3 F39 x13 �39x11 C 468x9 �1989x7 �507x6 C 2886x5 C 1443x4 �624x3

�234x2 C 3

T2 D13 x13 �2x12 C 4x10 �5x9 C x8 C 5x7 �11x6 C 19x5 �22x4 C 16x3

�10x2 C 6x �1

T1 13 x13 �x12 �24x11 C 19x10 C 190x9 �116x8 �601x7 C 246x6 C 738x5

�215x4 �291x3 C 68x2 C 10x �1



S14 x14 �x �1

A14 x14 �9x7 C 49x5 �90

T61 x14 C x2 �2x C 1

T60 x14 �7x8 �6x7 C 49x2 C 84x C 36

T59 x14 �96x7 �1568x2 C 2304

T58 x14 �7x8 C 6x7 C 784x2 �1344x C 576

T57 x14 �x2 C 1

T56 x14 C 14x8 �24

T55 x14 �x2 �1

T54 x14 C 7x6 C 4

T53 x14 C 7x8 �7x6 �9

T52 x14 �x13 C x12 �x11 C x10 �4x8 C 6x7 �5x5 C 5x4 C x3 �4x2 C 1

T51 x14 �7x2 �3

T50 x14 �2x8 �5x6 �3x2 �4

T49 x14 �4x6 C 4

T48 x14 C 7x6 C 21x2 C 50

T47 x14 C 2x12 �2x10 C x6 �8x4 C 5x2 C 2

T46 x14 C 5x10 �4x8 C 2

T45 x14 �7x12 �14x11 C 21x10 C 84x9 C 35x8 �69x7 C 7x6 C 84x5 C 7x4

C77x3 C 133x2 C 35x C 58

T44 x14 �8x10 �2x8 C 16x6 C 6x4 �6x2 �2

T43 x14 C 3x12 �4x8 C x6 �3x2 C 1

T42 x14 C 7x12 �7x10 �49x8 C 7x6 C 49x4 �49x2 C 9

T41 x14 �2x12 �2x10 C x8 C 6x6 �x2 �4

T40 x14 C 2x12 �14x8 C 35x6 �21x4 �7x2 C 7

PGL2.13/ x14 �x13 �26x10 C 65x6 C 13x5 �52x2 �12x �1

T38 x14 �7x8 �14x6 �7

T37 x14 �28x11 �28x9 C 196x8 �2x7 C 392x6 C 616x4 �392x3

C14x2 C 56x C 9

T36 x14 �35x12 �133x11 C 469x10 C 1239x9 C 742x8 �3604x7 C 47138x6

�85351x5 C 168028x4 �156394x3 C 158718x2 �72149x C 42751

T35 x14 �9x12 C 17x10 C 29x8 �49x6 �67x4 �21x2 �1

T34 x14 �3x12 C 4x8 C x6 �3x2 �1

T33 x14 C 14x10 C 28x8 �35x6 C 784x4 �140x2 �4

T32 x14 �14x12 C 77x10 �210x8 C x7 C 294x6 �7x5 �196x4 C 14x3

C49x2 �7x C 2

T31 x14 �7x12 C 91x8 �192x7 �126x5 �1519x4 C 1218x3 C 8827x2

C11046x C 5484

L2.13/ x14 �6x13 C 13x12 �338x9 C 845x8 C 17576x4 C 70304x C 35152

T29 x14 C 12x12 C 41x10 C 26x8 �59x6 �64x4 C 9x2 C 17

T28 x14 C 7x6 C 7x4 C 7x2 �1

T27 x14 C 7x8 �14x6 C 7

T26 x14 �28x11 C 280x10 C 567x9 C 5061x8 C 2273x7 �735x6 C 33908x5

C40348x4 �3192x3 C 36855x2 C 119196x C 75141


T25 x14 C 42x12 �42x11 C 525x10 �896x9 C 2422x8 �2536x7 C 1225x6

C742x5 �994x4 C 560x3 �28x2 �168x C 56

T24 x14 �3x7 C 6

T23 x14 �14x12 C 77x10 �210x8 �11x7 C 294x6 C 77x5 �196x4 �154x3 C 49x2

C77x C 29

T22 x14 C 42x12 �840x11 C 4473x10 �77728x9 C 235648x8 �2601696x7

C6832756x6 �48638016x5 C 124211584x4 �490172256x3

C802837840x2 �1497646080x C 723639232

T21 x14 �x12 �12x10 C 7x8 C 28x6 �14x4 �9x2 �1

T20 x14 �2x13 �4x12 C x11 C 6x9 C 10x8 �x7 C 6x6 �13x4 �15x3 �5x2 C x �1

T19 x14 C 10x8 C 8x6 �4x4 C 2

T18 x14 C 4x12 �30x10 C 8x8 C 60x6 C 8x4 �24x2 �8

T17 x14 C 11x12 C 53x10 C 15x8 �149x6 C 89x4 �x2 �3

T16 x14 �14x10 C 14x8 C 22x7 C 21x6 C 49x4 �154x3 C 77x2 �154x C 149

T15 x14 �87x12 C 1456x10 �256x9 �8563x8 C 3448x7 C 18032x6 �9890x5

�11776x4 C 5198x3 C 3128x2 �506x �184

T14 x14 �2x7 C 8

T13 x14 C 4x13 C 10x11 C 39x10 C 28x9 �13x8 C 34x7 C 126x6 �36x5 C 29x4

�24x3 C 38x2 �16x C 4

T12 x14 C 35x12 C 210x11 C 735x10 C 2849x9 C 10150x8 C 45655x7 C 94570x6

C98455x5 �199381x4 �344400x3 C 647395x2 C 4094650x C 1010645

T11 x14 �5x12 �11x10 C 25x8 C 27x6 �23x4 �17x2 �1

T10 x14 C 14x8 �84x6 C 84x4 C 21x2 �9

T9 x14 C 7x12 �49x10 �245x8 C 588x6 C 294x4 �7

T8 x14 �x12 �3x11 C 5x10 C 5x9 �5x8 �9x7 C x6 C 14x5 �2x4 �7x3 C x2 C 1

T7 x14 C 2

T6 x14 C 13x12 C 31x10 �9x8 �54x6 �3x4 C 23x2 �1

T5 x14 �x7 C 2

T4 x14 C 7

T3 x14 C 6x12 C 7x10 C x8 �3x6 C x4 C 3x2 C 1

T2 x14 C 8x12 C 22x10 C 8x8 �55x6 �48x4 C 64x2 C 71

T1 x14 C 25x12 C 214x10 C 767x8 C 1194x6 C 686x4 C 53x2 C 1

References

Abhyankar, S.S. (1957): Coverings of algebraic curves. Amer. J. Math. 79, 825–856Abhyankar, S.S. (1994): Nice equations for nice groups. Israel J. Math. 88, 1–23Abhyankar, 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, 47–72Artin, E. (1947): Theory of braids. Ann. of Math. 48, 101–126Artin, 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


515

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 coveringsof curves. J. Algebra 125, 236–255

Beckmann, S. (1991): On extensions of number fields obtained by specializingbranched 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 givenclassical Galois group. J. reine angew. Math. 341, 147–156

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

Beynon, W.M., Spaltenstein, N. (1984): Green functions of finite Chevalley groupsof type En (n D 6;7;8). J. Algebra 88, 584–614

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

Borel, A. (1991): Linear algebraic groups. Springer, New YorkBorel, A. et al. (1970): Seminar on algebraic groups and related finite groups. LNM131, Springer, Berlin

Bosch, S., Guntzer, U., Remmert, R. (1984): Non-archimedean analysis. Springer,Berlin Heidelberg New York

Breuer, T., Malle, G., O’Brien, E. (2016): Reliability and reproducibility of Atlasinformation. 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 ofprime 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, ChichesterChang, B., Ree, R. (1974): The characters of G2.q/. Pp. 395–413 in: SymposiaMathematica XIII, London

Chow, W.-L. (1948): On the algebraic braid group. Ann. of Math. 49, 654–658Cohen, A.M. et al. (1992): The local maximal subgroups of exceptional groups ofLie type, finite and algebraic. Proc. London Math. Soc. 64, 21–48

Cohen, D.B. (1974): The Hurwitz monodromy group. J. Algebra 32, 501–517Colin, 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, OxfordCooperstein, B.N. (1981): Maximal subgroups of G2.2n/. J. Algebra 70, 23–36Crespo, T. (1989): Explicit construction of QAn type fields. J. Algebra 127, 452–461Debes, 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 finitefields. Ann. of Math. 103, 103–161

Delone, B.N., Faddeev, D.K. (1944): Investigations in the geometry of the Galoistheory. 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)]

Dentzer, R. (1989): Projektive symplektische Gruppen PSp4.p/ als Galoisgruppenuber 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 Chevalleygroups E7 and E8. Tokyo J. Math. 6, 191–216

Deriziotis, D.I., Michler, G.O. (1987): Character table and blocks of finite simpletriality groups 3D4.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 theinverse 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. LMSStudent 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 auxgroupes 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. Paris258, 5305–5308

Elkies, N.D. (1997): Linearized algebra and finite groups of Lie type I: Linear andsymmetric groups. Contemp. Math. 245, 77–108

Engler, A.J., Prestel, A. (2005): Valued fields. Springer Verlag, Berlin

518 References

Enomoto, H. (1976): The characters of the finite Chevalley groups G2.q/, q D 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 onevariable. Trudy Mat. Inst. Steklov 38, 190–243 (Russian) [English transl.: Amer.Math. Soc. Transl. II 3, 15–38 (1956)]

Fadell, E. (1962): Homotopy groups of configuration spaces and the string problemof Dirac. Duke Math. J. 29, 231–242

Fadell, E., Van Buskirk, J. (1962): The braid groups of E2 and S2. Duke Math. J.29, 243–257

Feit, W. (1984): Rigidity of Aut.PSL2.p2//, p � ˙2 .mod 5/, p ¤ 2. Pp. 351–356in: 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 finitegroups of Lie type E6 and E7. Comm. Algebra 21, 93–161

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

Forster, O. (1981): Lectures on Riemann surfaces. Springer, New YorkFranz, 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

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, BerlinFried, M.D., Volklein, H. (1991): The inverse Galois problem and rational points onmoduli 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-Whitneyclasses and Hasse-Witt invariants. J. reine angew. Math. 360, 84–123

Frohardt, D., Guralnick, R., Hoffman, C., Magaard, K. (2016): Exceptional lowgenus 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 additivepolynomials. 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 groupefondamental en geometrie algebrique. Birkhauser, Basel

Gillette, R., Van Buskirk, J. (1968): The word problem and consequences for thebraid 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, BerlinGranboulan, L. (1996): Construction d’une extension reguliere de Q.T / de groupede Galois M24. Experimental Math. 5, 3–14

Grothendieck, A. (1971): Revetements etales et groupe fondamental. Springer,Berlin

Guralnick, R.M., Lubeck, 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. Algebra131, 303–341

Hafner, F. (1992): Einige orthogonale und symplektische Gruppen als Galoisgrup-pen uber Q. Math. Ann. 292, 587–618

Hansen, V.L. (1989): Braids and coverings. CambridgeUniversity Press, CambridgeHaraoka, Y. (1994): Finite monodromy of Pochhammer equation. Ann. Inst. Fourier44, 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, 281–293Harbater, 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., Zurich 1994 , Birkhauser

Hartshorne, R. (1977): Algebraic geometry. Springer, Berlin Heidelberg New YorkHasse, H. (1948): Existenz und Mannigfaltigkeit abelscher Algebren mit vorgege-bener Galoisgruppe uber einem Teilkorper des Grundkorpers I. Math. Nachr. 1,40–61

Hasse, H. (1980): Number theory. Springer, BerlinHecke, E. (1935): Die eindeutige Bestimmung der Modulfunktionen q-ter Stufedurch 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, NewYork

Hoechsmann, K. (1968): Zum Einbettungsproblem. J. reine angew. Math. 229,81–106

Hoyden-Siedersleben, G. (1985): Realisierung der Jankogruppen J1 und J2 alsGaloisgruppen uber Q. J. Algebra 97, 14–22

Hoyden-Siedersleben, G., Matzat, B.H. (1986): Realisierung sporadischer einfacherGruppen als Galoisgruppen uber 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, BerlinHuppert, 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. 99–120in: Proc. Int. Congress of Math., Kyoto 1990, Springer

Ihara, Y. et al. (eds.) (1989): Galois groups over Q. Springer, New YorkIitaka, S. (1982): Algebraic geometry. Springer Verlag, BerlinIkeda, M. (1960): Zur Existenz eigentlicher galoisscher Korper beim Einbettungs-problem fur 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 inGalois theory. Amer. Math. Soc., Providence

Iwasawa, K. (1953): On solvable extensions of algebraic number fields. Ann. ofMath. 58, 548–572

References 521

Jacobson, N. (1980): Basic algebra II. W. H. Freeman, New YorkJensen, C., Ledet, A., Yui, N. (2002): Generic polynomials. Constructive aspects ofthe 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., Lubeck, F., Magaard, K. (2001): Matrix generators for the Ree groups2G2.q/. Comm. Algebra 29, 407–415

Kleidman, P.B. (1987): The maximal subgroups of the finite 8-dimensional orthog-onal groups P�C

8 .q/ and of their automorphism groups. J. Algebra 110, 173–242Kleidman, P.B. (1988a): The maximal subgroups of the Steinberg triality groups

3D4.q/ and of their automorphism groups. J. Algebra 115, 182–199Kleidman, P.B. (1988b): The maximal subgroups of the Chevalley groups G2.q/

with q odd, the Ree groups 2G2.q/ and of their automorphism groups. J. Algebra117, 30–71

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

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

Kluners, J., Malle, G. (2000): Explicit Galois realization of transitive groups ofdegree up to 15. J. Symbolic Comput. 30, 675–716

Kluners, J., Malle, G. (2002): A Database for Number Fields. http://galoisdb.math.upb.de/

Kochendorffer, R. (1953): Zwei Reduktionssatze zum Einbettungsproblem fur abel-sche Algebren. Math. Nachr. 10, 75–84

Konig, J. (2015): Computation of Hurwitz spaces and new explicit polynomials foralmost simple Galois groups. Math. Comp., to appear

Konig, J. (2016): On rational functions with monodromy group M11. J. SymbolicComput. 79, 372–383

Kopf, U. (1974): Uber eigentliche Familien algebraischer Varietaten uber affinoidenRaumen. Schriftenreihe Math. Inst. Univ. Munster, 2. Serie, Heft 7

Kreuzer, M., Robbiano, L. (2000): Computational commutative algebra I. SpringerVerlag, Berlin

Krull, W., Neukirch, J. (1971): Die Struktur der absoluten Galoisgruppe uber demKorper 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, 112–124Landazuri, 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, NewYork

http://galoisdb.math.upb.de/

http://galoisdb.math.upb.de/

522 References

Ledet, A. (2000): On a theorem of Serre. Proc. Amer. Math. Soc. 128, 27–29Ledet, A. (2005): Brauer type embedding problems. Amer. Math. Soc., ProvidenceLiebeck, M.W., Seitz, G.M. (1999): On finite subgroups of exceptional algebraicgroups. 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)

Lubeck, F., Malle, G. (1998): .2;3/-generation of exceptional groups. J. LondonMath. Soc. 59, 109–122

Lusztig, G. (1977): Representations of finite Chevalley groups. CBMS RegionalConference Series in Mathematics 39

Lusztig, G. (1980): On the unipotent characters of the exceptional groups over finitefields. Invent. Math. 60, 173–192

Lusztig, G. (1984): Characters of reductive groups over a finite field. Annals ofMath. Studies 107, Princeton University Press, Princeton

Lyndon, R.C., Schupp, P.E. (1977): Combinatorial group theory. Springer, BerlinMadan, M., Rzedowski-Calderon, M., Villa-Salvador, G. (1996): Galois extensionswith 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 ofdegree d � 15. J. Symb. Comp. 4, 83–92

Malle, G. (1988a): Polynomials with Galois groups Aut.M22/, M22, andPSL3.IF4/: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, 64–97Malle, 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 groupsG2.q/ as Galois groups.Pacific J. Math. 212, 157–167

Malle, G., Matzat, B. H. (1985): Realisierung von Gruppen PSL2.IFp/ als Galois-gruppen uber 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 Galoisgroups over Qab.t/. Israel J. Math. 96, 431–444.

Malle, G., Testerman, D. (2011): Linear algebraic groups and finite groups of Lietype. Cambridge University Press, Cambridge

Matsumura, H. (1980): Commutative algebra. Benjamin/Cummings, ReadingMatzat, B.H. (1979): Konstruktion von Zahlkorpern mit der GaloisgruppeM11 uberQ.

p�11/. Manuscripta Math. 27, 103–111Matzat, 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 Zahlentheoriemit 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, BerlinMatzat, 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 uber 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 modularforms. Kluwer Acad. Publ., Boston, MA

Matzat, B.H., Zeh-Marschke, A. (1986): Realisierung der MathieugruppenM11 undM12 als Galoisgruppen uber Q. J. Number Theory 23, 195–202

McLaughlin, J. (1969): Some subgroups of SLn.IFp/. Ill. J. Math. 13, 108–115Melnicov, O.V. (1980): Projective limits of free profinite groups. Dokl. Akad. NaukSSSR 24, 968–970

Mestre, J.-F. (1990): Extensions regulieres de Q.T / de groupe de Galois QAn. J.Algebra 131, 483–495

Mestre, J.-F. (1994): Construction d’extensions regulieres de Q.t/ a groupes deGalois SL2.F7/ et QM12. C. R. Acad. Sci. 319, 781–782

Mestre, J.-F. (1998): Extensions Q-regulieres de Q.t/ de groups de Galois 6:A6 et6:A7. Israel J. Math. 107, 333–341

524 References

Michler, G.O., Ringel, C.M. (eds.) (1991): Representation theory of finite groupsand 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 Chevalleygroups E7 and E8. Tokyo J. Math. 3, 391–461

Monien, H. (2017): The sporadic group J2, Hauptmodul and Belyi map. Preprint,arXiv:1703.05200

Muller, P. (2012): A one-parameter family of polynomials with Galois group M24

over Q.t/. Preprint, arXiv:1204.1328Nagata, M. (1977): Field theory. Marcel Dekker, New YorkNakamura, H. (1997): Galois rigidity of profinite fundamental groups. SugakuExpositiones 10, 195–215

Narkiewicz, W. (1990): Elementary and analytic theory of algebraic numbers.Springer, Berlin

Neukirch, J., Schmidt, A. andWingberg, 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

Norton, S.P., Wilson, R.A. (1989): The maximal subgroups of F4.2n/ 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]

Pochhammer, L. (1870): Uber die hypergeometrischen Funktionen n-ter Ordnung.J. reine angew. Math. 71, 312–352

Pop, F. (1994): 12Riemann existence theorem with Galois action. Pp. 193–218 in:

G. Frey and J. Ritter (eds.): Algebra and number theory. W. de Gruyter, BerlinPop, 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, 1–34Popp, H. (1970): Fundamentalgruppen algebraischer Mannigfaltigkeiten. Springer,Berlin

Porsch, U. (1993): Greenfunktionen der endlichen Gruppen E6.q/, q D 3n. 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. SpringerVerlag, Berlin

References 525

Raynaud, M. (1994): Revetements de la droite affine en caracteristique p > 0 etconjecture d’Abhyankar. Invent. Math. 116, 425–462

Reichardt, H. (1937): Konstruktion von Zahlkorpern mit gegebener Galoisgruppevon 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’sUniversity, Kingston

Rzedowski-Calderon, M. (1989): Construction of global function fields with nilpo-tent automorphism groups. Bol. Soc. Mat. Mexicana 34, 1–10

Safarevic, 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)]

Safarevic, I.R. (1954a): On the construction of fields with a given Galois group oforder `a. Izv. Akad. Nauk. SSSR 18, 216–296 (Russian) [English transl.: Amer.Math. Soc. Transl. II 4, 107–142 (1956)]

Safarevic, 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)]

Safarevic, I.R. (1954c): On the problem of imbedding fields. Izv. Akad. Nauk. SSSR18, 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 givensolvable Galois group. Izv. Akad. Nauk. SSSR 18, 525–578 (Russian) [Englishtransl.: Amer. Math. Soc. Transl. II 4, 185–237 (1956)]

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

Safarevic, I.R. (1989): Factors of descending central series. Math. Notes 45, 262–264

Safarevic, I.R. (1994): Basic algebraic geometry I. Springer VerlagSaıdi, 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 engeometrie 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, BerlinSchneps, L. (1992): Explicit construction of extensions of Q.t/ of Galois group QAn

for n odd. J. Algebra 146, 117–123Scholz, A. (1929): Uber die Bildung algebraischer Zahlkorper mit auflosbarergaloisscher Gruppe. Math. Z. 30, 332–356

Scholz, A. (1937): Konstruktion algebraischer Zahlkorper mit beliebiger Gruppevon Primzahlpotenzordnung I. Math. Z. 42, 161-188

526 References

Scott, L. (1977): Matrices and cohomology. Ann. of Math. 105, 473–492Seidelmann, 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, LeipzigSerre, J.-P. (1956): Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier6, 1–42

Serre, J.-P. (1959): Groupes algebriques et corps de classes. Hermann, ParisSerre, J.-P. (1964): Cohomologie galoisienne. Springer, BerlinSerre, J.-P. (1979): Local fields. Springer, New YorkSerre, 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, 73–85Serre, 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, BostonShabat, 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 UniversityPress, Princeton

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

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

Shiina, Y. (2003a): Braid orbits related to PSL2.p2/ and some simple groups.Tohoku Math. J. 55, 271–282

Shiina, Y. (2003b): Regular Galois realizations of PSL2.p2/ 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/ overfinite 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 finitefields 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 YorkSimpson, 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, WellesleySonn, 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 ofQ.T /. J. Algebra 140, 355–359

Sonn, J. (1994a): Brauer groups, embedding problems, and nilpotent groups asGalois groups. Israel J. Math. 85, 391–405

Sonn, J. (1994b): Rigidity and embedding problems over Qab.t/. J. Number Theory47, 398–404

Spaltenstein, N. (1982): Caracteres unipotents de 3D4.IFq/. 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, BaselSteinberg, R. (1965): Regular elements of semi-simple algebraic groups. Publ.Math. IHES 25, 49–80.

Steinberg, R. (1967): Lectures on Chevalley groups. Yale UniversitySteinberg, 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, StuttgartStoll, 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, BerlinSuzuki, M. (1962): On a class of doubly transitive groups. Ann. of Math. 75,105–145

Suzuki, M. (1982): Group theory. Springer, BerlinSwallow, J.R. (1994): On constructing fields corresponding to the QAn’s of Mestrefor 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 numberfields. 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.�n/. J. Algebra 89, 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: TuanHsio-Fu (ed.): Group theory, Beijing 1984. Springer, Berlin

Thompson, J.G., Volklein, H. (1998): Symplectic groups as Galois groups. J. GroupTheory 1, 1–58

528 References

Tschebotarow, N.G., Schwerdtfeger, H. (1950): Grundzuge der Galoisschen Theo-rie. Noordhoff, Groningen

Uchida, K. (1980): Separably Hilbertian fields. Kodai Math. J. 3, 83–95Vila, 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, 144–152Volklein, 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 groupsPGLn.q/ and PUn.q2/. 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, 421–438Wagner, A. (1978): Collineation groups generated by homologies of order greaterthan 2. Geom. Dedicata 7, 387–398

Wagner, A. (1980): Determination of the finite primitive reflection groups over anarbitrary 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 YorkWehrfritz, B.A.F. (1973): Infinite linear groups. Springer Verlag, BerlinWeigel, T. (1992): Generators of exceptional groups of Lie type. Geom. Dedicata41, 63–87

Weil, A. (1956): The field of definition of a variety. Amer. J. Math. 78, 509–524Weil, A. (1974): Basic number theory. Springer, New YorkWeissauer, R. (1982): Der Hilbertsche Irreduzibilitatssatz. J. reine angew. Math.334, 203–220

Wilson, R.A. (2017): Maximal subgroups of sporadic groups. To appear in: FiniteSimple Groups: Thirty Years of the Atlas and Beyond, AMS ContemporaryMath-ematics Series, 2017.

Wolf, P. (1956): Algebraische Theorie der galoisschen Algebren. VEB DeutscherVerlag der Wissenschaften, Berlin

Yakovlev, A.V. (1964): The embedding problems for fields. Izv. Akad. Nauk. SSSR28, 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)]

Zalesskiı, A.E., Serezkin, 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, GottingenZywina, D (2015): The inverse Galois problem for PSL2.IFp/. Duke Math. J. 164,2253–2292.

Index

�-equivalent matrix, 386

accompanying Brauer embedding problem,350

accompanying embedding problem, 350admissible covering, 451admissible subset, 451affinoid analytic space, 451algebraic fundamental group, 4, 187almost character, 127arithmetic fundamental group, 10, 197(full) Artin braid group, 179associated F-module, 390AV -rigid, 64AV -symmetric, 64AV -symmetrized irrationality degree, 64

basic rigidity theorem, 30Belyi triple, 102braid cycle theorem, 246braid orbit theorem, 215braid relations, 181Brauer embedding problem, 339

central embedding problem, 288characteristic polynomial of an F-module, 389clean Belyi function, 17closed ultrametric disc, 462coherent sheaf, 454cohomologically trivial in dimension i , 361companion matrix, 390comparison theorem of Tate, 367compatible family, 467concordance obstruction, 364concordant embedding problem, 351

conformal orthogonal group, 111conformal symplectic group, 108connected rigid analytic space, 452convolution, 251coroot, 93cyclic F-module, 385cyclotomic character, 14cyclotomic polynomial, 117

Dedekind criterion, 72Dickson algebra, 396Dickson invariants, 395Dickson polynomial, 395disclosed function field of one variable, 14duality theorem of Tate, 354dualizable F-module, 385

effective G-module, 392embedding problem, 288existentially closed, 474extension theorem, 67

F-field, 385F-module, 385field of definition, 19field of definition with group, 19field of invariants, 394field of moduli, 30field restriction of algebraic groups, 441finite embedding problem, 288finite morphism, 458first embedding obstruction, 364fixed point theorem, 53Frattini embedding problem, 288Frattini embedding theorem, 319Freiheitssatz of Iwasawa, 295


531

https://doi.org/10.1007/978-3-662-55420-3

532 Index

Frobenius endomorphism, 385Frobenius field, 385Frobenius module, 385full symmetry group, 31, 63fundamental solution matrix, 387fundamental system of solutions, 387

G-compatible family, 469G-realization, 34G-relative H -invariant, 396G-relative Colin Matrix, 399G-relative resolvent, 396GA-realization, 36GAGA for IP1, 456Galois group of an F-module, 390GAR-realization, 302general unitary group, 107generating s-system, 26generic polynomial, 396geometric (proper) solution, 289geometric embedding problem, 289geometric field extension, 8geometrically conjugate, 126GL-stable tuple, 260gluing datum, 451gluing of morphisms, 452gluing of spaces, 451good reduction modulo p, 88Green function, 128group of geometric automorphisms, 43

HVs -rigid class vector, 212

Hasse embedding obstruction, 364Hasse-Witt-invariant, 332Hilbertian field, 287Hilbertian set, 287homology, 100homomorphism ramified in, 480(full) Hurwitz braid group, 181Hurwitz classification, 27, 198hypothesis (H), 254

induced cover, 461irrationality degree, 28irreducible Jordan–Pochhammer tuple, 270

j -th braid orbit genus, 213Jordan–Pochhammer tuple, 270

k-rational class vector, 319k-symmetric class vector, 319kernel of an embedding problem, 288

large field, 475Lemma of Scott, 260Lemma of Speiser, 201linear Tschirnhaus transform, 401linearly rigid tuple, 260Lusztig series, 126

M-section, 469mapping class group, 183modular Dedekind criterion, 403modular Galois theory, 383Moore determinant, 387Moore matrix, 387morphism of rigid analytic spaces, 451multiplication with c, 267

non-split embedding problem, 288normalized structure constant, 36

open ultrametric disc, 462orthogonal group, 110orthogonal group of minus type, 115orthogonal group of plus type, 111

}-stable, 86pairwise adjusted, 471Pochhammer transform, 251Pochhammer transformation, 251primitive linear group, 100primitive prime divisor, 118primitive translate, 55profinite Hurwitz braid group, 189profinite Riemann existence theorem, 4projective profinite group, 294proper solution (field) of an embedding

problem, 288pseudo algebraically closed, 229pseudo Steinberg cross section, 424pseudo-reflection, 100pure Artin braid group, 179pure Hurwitz braid group, 181

q-additive polynomial, 388quasi-central element, 132quasi-determinant, 115quasi-p-group, 484

r-fold uncomplete product, 179r-fold uncomplete symmetric product, 179rational class vector, 29rational subset, 450rationally rigid class vector, 29reduced braid orbit genera, 245

Index 533

reflection, 100regular solution of an embedding problem, 289regularity theorem, 212relative Reynolds operator, 398rigid analytic space, 451rigid braid cycle, 247rigid braid cycle theorem, 247rigid braid orbit theorem, 216rigid class vector, 29rigid HV

S -orbit, 48, 64rigid HV

s -orbit, 212rigidity defect, 263ring of holomorphic functions, 450ring of invariants, 394robust generating systems, 407root, 93

s-th V -symmetrized braid orbit genus, 235Scholz embedding problem, 374Scholz extension, 374Scholz solution, 374Schur multiplier, 226second embedding obstruction, 364semiabelian group, 299semirational class, 41shape function, 226socle of an `-Galois extension, 375solution field of an embedding problem, 288solution field of an F-module, 385solution of an embedding problem, 288solution space of an F-module, 385specialization theorem, 224sphere relations, 182spinor norm, 110split embedding problem, 288splitting theorem, 13, 196stability condition, 260

Steinberg cross section, 406Steinberg endomorphism, 423strictly non-degenerate quadratic form, 332strong rigidity theorem, 32symmetric algebra, 394symmetry group, 31, 63symplectic group, 108

Tate algebra, 449thick normal subgroup, 185transference, 347translation theorem, 58transvection, 100trivial cover, 461trivial F-module, 385twisted braid orbit theorem, 239twisted rigidity theorem, 50twisted structure sheaf, 455twisted upper bound theorem, 423

uniform function, 126unipotent character, 127uniquely liftable, 318unirational function field, 200universally central embeddable Galois

extension, 328unramified, 187unramified rational place, 223upper bound theorem, 391

V -configuration, 48, 232V -rigid class vector, 48V -symmetric, 31V -symmetrized braid orbit, 211V -symmetrized irrationality degree, 31

wreath extension, 347

Documents

Appendix: Example Polynomials