Galois groups of tamely ramified p-extensions
Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 59-70.

On connait très peu à propos du groupe de Galois de la p-extension maximale non-ramifiée en dehors d’un ensemble fini S de nombres premiers d’un corps de nombres lorsque les nombres premiers au-dessus de p ne sont pas dans S. Nous décrivons des méthodes pour calculer ce groupe quand il est fini et ses propriétées conjecturales quand il est infini.

Very little is known regarding the Galois group of the maximal p-extension unramified outside a finite set of primes S of a number field in the case that the primes above p are not in S. We describe methods to compute this group when it is finite and conjectural properties of it when it is infinite.

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DOI : https://doi.org/10.5802/jtnb.573
@article{JTNB_2007__19_1_59_0,
     author = {Nigel Boston},
     title = {Galois groups of tamely ramified $ p$-extensions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {59--70},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.573},
     zbl = {1123.11038},
     mrnumber = {2332053},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.573/}
}
Nigel Boston. Galois groups of tamely ramified $ p$-extensions. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 59-70. doi : 10.5802/jtnb.573. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.573/

[1] M.Abért, B.Virág, Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc. 18 (2005), 157–192. | MR 2114819 | Zbl 02125234

[2] W.Aitken, F.Hajir, C.Maire, Finitely ramified iterated extensions. Int. Math. Res. Not. 14 (2005), 855–880. | MR 2146860 | Zbl 02188805

[3] Y.Barnea, M.Larsen, A non-abelian free pro-p group is not linear over a local field. J. Algebra 214 (1999), 338–341. | MR 1684856 | Zbl 0923.20018

[4] Y.Barnea, A.Shalev, Hausdorff dimension, pro-p groups, and Kac-Moody algebras. Trans. Amer. Math. Soc. 349 (1997), 5073–5091. | MR 1422889 | Zbl 0892.20020

[5] L.Bartholdi, M.R.Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL 2 (Z 3 ). To appear in J. Number Theory.

[6] L.Bartholdi, B.Virág, Amenability via random walks. To appear in Duke Math. J. | MR 2176547 | Zbl 1104.43002

[7] W.Bosma, J.Cannon, Handbook of MAGMA Functions. Sydney: School of Mathematics and Statistics, University of Sydney, 1993.

[8] N.Boston, Some Cases of the Fontaine-Mazur Conjecture II. J. Number Theory 75 (1999), 161–169. | MR 1681626 | Zbl 0928.11050

[9] N.Boston, Reducing the Fontaine-Mazur conjecture to group theory. Progress in Galois theory (2005), 39–50. | MR 2148459 | Zbl 02163150

[10] N.Boston, Embedding 2-groups in groups generated by involutions. J. Algebra 300 (2006), 73–76. | MR 2228635 | Zbl 1102.12002

[11] N.Boston, C.R.Leedham-Green, Explicit computation of Galois p-groups unramified at p. J. Algebra 256 (2002), 402–413. | MR 1939112 | Zbl 1016.11051

[12] N.Boston, C.R.Leedham-Green, Counterexamples to a conjecture of Lemmermeyer. Arch. Math. Basel 72 (1999), 177–179. | MR 1671275 | Zbl 0922.11095

[13] D.J.Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory. J. Math. Phys. (to appear).

[14] M.R.Bush, Computation of Galois groups associated to the 2-class towers of some quadratic fields. J. Number Theory 100 (2003), 313–325. | MR 1978459 | Zbl 1039.11091

[15] H. Cohen, H. W. Lenstra, Jr., Heuristics on class groups. Lecture Notes in Math. 1086, Springer-Verlag, Berlin 1984. | MR 750661 | Zbl 0532.12008

[16] H.Cohn, J.C.Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(dp) as p varies. Math. Comp. 37 (1983), 711–730. | MR 717716 | Zbl 0523.12002

[17] B.Eick, H.Koch, On maximal 2-extensions of Q with given ramification. Proc. St. Petersburg Math. Soc. (Russian), American Math. Soc. Translations (English) (to appear). | MR 2276852

[18] J.-M.Fontaine, B.Mazur, Geometric Galois representations. Proceedings of a conference held in Hong Kong, December 18-21, 1993,” International Press, Cambridge, MA and Hong Kong. | Zbl 0839.14011

[19] J.Gilbey, Permutation groups, a related algebra and a conjecture of Cameron. Journal of Algebraic Combinatorics, 19 (2004) 25–45. | MR 2056765 | Zbl 1080.20002

[20] E.S.Golod, I.R.Shafarevich, On class field towers (Russian). Izv. Akad. Nauk. SSSR 28 (1964), 261–272. English translation in AMS Trans. (2) 48, 91–102. | MR 161852 | Zbl 0148.28101

[21] R.Grigorchuk, Just infinite branch groups. New Horizons in pro-p Groups, Birkhauser, Boston 2000. | MR 1765119 | Zbl 0982.20024

[22] R.Grigorchuk, A.Zuk, On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput., 12 (2000), 223–246. | MR 1902367 | Zbl 1070.20031

[23] F.Hajir, C.Maire, Tamely ramified towers and discriminant bounds for number fields. II. J. Symbolic Comput. 33 (2002), 415–423. | MR 1890578 | Zbl 1086.11051

[24] G.Havas, M.F.Newman, E.A.O’Brien, Groups of deficiency zero. Geometric and Computational Perspectives on Infinite Groups, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25 (1996) 53–67. | Zbl 0849.20019

[25] H.Kisilevsky, Number fields with class number congruent to 4(mod8) and Hilbert’s theorem 94. J. Number Theory 8 (1976), no. 3, 271–279 | Zbl 0334.12019

[26] H.Koch, Galois theory of p-extensions. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002. | MR 1930372 | Zbl 1023.11002

[27] H.Koch, B.Venkov, The p-tower of class fields for an imaginary quadratic field (Russian). Zap. Nau. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 46 (1974), 5–13. | MR 382235 | Zbl 0335.12022

[28] J.Labute, Mild pro-p-groups and Galois groups of p-extensions of Q. J. Reine Angew. Math. (to appear). | MR 2254811 | Zbl 05080511

[29] A.Lubotzky, Group presentations, p-adic analytic groups and lattices in SL 2 (C). Ann. Math. 118 (1983), 115–130. | MR 707163 | Zbl 0541.20020

[30] J.Mennicke, Einige endliche Gruppe mit drei Erzeugenden und drei Relationen. Arch. Math. X (1959), 409–418. | MR 113946 | Zbl 0089.01405

[31] E.A.O’Brien, The p-group generation algorithm. J. Symbolic Comput. 9 (1990), 677–698. | Zbl 0736.20001

[32] R.W.K.Odoni, Realising wreath products of cyclic groups as Galois groups. Mathematika 35 (1988), 101–113. | MR 962740 | Zbl 0662.12010

[33] I.R.Shafarevich, Extensions with prescribed ramification points (Russian). IHES Publ. Math. 18 (1964), 71–95. | Numdam | MR 176979

[34] M.Stoll, Galois groups over Q of some iterated polynomials. Arch. Math. (Basel) 59 (1992), 239–244. | MR 1174401 | Zbl 0758.11045

[35] G.Willis, Totally disconnected, nilpotent, locally compact groups. Bull. Austral. Math. Soc. 55 (1997), 143–146. | MR 1428537 | Zbl 0893.22001

[36] E.Zelmanov, On groups satisfying the Golod-Shafarevich condition. New horizons in pro-p groups, Birkhaüser Boston, Boston, MA, 2000. | MR 1765122 | Zbl 0974.20022

[37] A.Zubkov, Non-abelian free pro-p-group are not represented by 2×2-matrices. Siberian Math.J, 28 (1987), 64–69. | Zbl 0653.20026