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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 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
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