We specialize various three-point covers to find number fields with Galois group , , , or and light ramification in various senses. One of our fields has the unusual property that it is ramified only at the single prime .
Par spécialisation de divers revêtements à trois points, on trouve des corps de nombres ayant groupe de Galois , , , ou et petite ramification selon divers aspects. Un de ces corps, de groupe de Galois , a la propriété remarquable de n’être ramifié qu’en .
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.948
Keywords: Number field, Discriminant, Ramification, Mathieu group
David P. Roberts 1
@article{JTNB_2016__28_2_435_0, author = {David P. Roberts}, title = {Lightly ramified number fields with {Galois} group $S.M_{12}.A$}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {435--460}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {28}, number = {2}, year = {2016}, doi = {10.5802/jtnb.948}, zbl = {1411.11102}, mrnumber = {3509719}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.948/} }
TY - JOUR AU - David P. Roberts TI - Lightly ramified number fields with Galois group $S.M_{12}.A$ JO - Journal de théorie des nombres de Bordeaux PY - 2016 SP - 435 EP - 460 VL - 28 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.948/ DO - 10.5802/jtnb.948 LA - en ID - JTNB_2016__28_2_435_0 ER -
%0 Journal Article %A David P. Roberts %T Lightly ramified number fields with Galois group $S.M_{12}.A$ %J Journal de théorie des nombres de Bordeaux %D 2016 %P 435-460 %V 28 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.948/ %R 10.5802/jtnb.948 %G en %F JTNB_2016__28_2_435_0
David P. Roberts. Lightly ramified number fields with Galois group $S.M_{12}.A$. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 435-460. doi : 10.5802/jtnb.948. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.948/
[1] P. Bayer, P. Llorente & N. Vila, « comme groupe de Galois sur », C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 7, p. 277-280. | Zbl
[2] W. Bosma, J. J. Cannon, C. Fieker & A. Steel (eds.), Handbook of Magma functions, 2.20 ed., University of Sydney Press, 2014.
[3] T. Breuer, « Multiplicity-Free Permutation Characters in GAP, part 2 », Manuscript (2006), 43 pages.
[4] H. Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000, xvi+578 pages. | DOI | Zbl
[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker & R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985, Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray, xxxiv+252 pages. | DOI | Zbl
[6] J. H. Conway & N. J. A. Sloane, Sphere packings, lattices and groups, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999, With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov, lxxiv+703 pages. | DOI
[7] L. Granboulan, « Construction d’une extension régulière de de groupe de Galois », Experiment. Math. 5 (1996), no. 1, p. 3-14. | DOI | Zbl
[8] J. W. Jones & D. P. Roberts, « A database of local fields », J. Symbolic Comput. 41 (2006), no. 1, p. 80-97, Database at http://math.la.asu.edu/~jj/localfields/. | DOI | MR | Zbl
[9] —, « Galois number fields with small root discriminant », J. Number Theory 122 (2007), no. 2, p. 379-407. | DOI | MR | Zbl
[10] —, « A database of number fields », LMS J. Comput. Math. 17 (2014), no. 1, p. 595-618, Database at http://hobbes.la.asu.edu/NFDB/. | DOI | MR | Zbl
[11] J. Klüners & G. Malle, « A database for field extensions of the rationals », LMS J. Comput. Math. 4 (2001), p. 182-196 (electronic), Database at http://galoisdb.math.upb.de/. | DOI | MR | Zbl
[12] G. Malle, « Polynomials with Galois groups and over », Math. Comp. 51 (1988), no. 184, p. 761-768. | DOI | Zbl
[13] G. Malle & B. H. Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999, xvi+436 pages. | DOI | MR | Zbl
[14] J. Martinet, « Petits discriminants des corps de nombres », in Number theory days, 1980 (Exeter, 1980), London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, p. 151-193. | DOI
[15] B. H. Matzat, « Konstruktion von Zahlkörpern mit der Galoisgruppe über », Arch. Math. (Basel) 40 (1983), no. 3, p. 245-254. | DOI | Zbl
[16] B. H. Matzat & A. Zeh-Marschke, « Realisierung der Mathieugruppen und als Galoisgruppen über », J. Number Theory 23 (1986), no. 2, p. 195-202. | DOI | Zbl
[17] J.-F. Mestre, « Construction d’extensions régulières de à groupes de Galois et », C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, p. 781-782. | Zbl
[18] P. Müller, « A one-parameter family of polynomials with Galois group over », , 2012. | arXiv
[19] B. Plans & N. Vila, « Galois covers of over with prescribed local or global behavior by specialization », J. Théor. Nombres Bordeaux 17 (2005), no. 1, p. 271-282. | DOI | MR | Zbl
[20] D. P. Roberts, « An construction of number fields », in Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, p. 237-267. | DOI | Zbl
[21] G. J. Schaeffer, « The Hecke stability method and ethereal modular forms », PhD Thesis, Berkeley (USA), 2012.
[22] The PARI group, Bordeaux, « PARI/GP », Version 2.3.4, 2009. | DOI | MR
Cited by Sources: