Lightly ramified number fields with Galois group $S.M_{12}.A$

David P. Roberts

doi:10.5802/jtnb.948

Lightly ramified number fields with Galois group

S . M_{12} . A

David P. Roberts ¹

¹ Division of Science and Mathematics University of Minnesota Morris Morris, MN 56267, USA

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 435-460.

Abstract
Résumé

We specialize various three-point covers to find number fields with Galois group $M_{12}$ , $M_{12} . 2$ , $2 . M_{12}$ , or $2 . M_{12} . 2$ and light ramification in various senses. One of our $2 . M_{12} . 2$ fields has the unusual property that it is ramified only at the single prime $11$ .

Par spécialisation de divers revêtements à trois points, on trouve des corps de nombres ayant groupe de Galois $M_{12}$ , $M_{12} . 2$ , $2 . M_{12}$ , ou $2 . M_{12} . 2$ et petite ramification selon divers aspects. Un de ces corps, de groupe de Galois $2 . M_{12} . 2$ , a la propriété remarquable de n’être ramifié qu’en $11$ .

Received: 2014-04-01
Revised: 2014-08-25
Accepted: 2014-09-04
Published online: 2017-01-02

MR: 3509719 | Zbl: 1411.11102

DOI: 10.5802/jtnb.948

Classification: 11R21, 11R29, 11R32, 11G32
Keywords: Number field, Discriminant, Ramification, Mathieu group

Author's affiliations:

David P. Roberts ¹

¹ Division of Science and Mathematics University of Minnesota Morris Morris, MN 56267, USA

@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/
UR  - https://zbmath.org/?q=an%3A1411.11102
UR  - https://www.ams.org/mathscinet-getitem?mr=3509719
UR  - https://doi.org/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://doi.org/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/

References
Cited by

[1] P. Bayer, P. Llorente & N. Vila, « ${\tilde{M}}_{12}$ comme groupe de Galois sur $Q$ », 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 $Q (T)$ de groupe de Galois $M_{24}$ », 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 $Aut (M_{22}), M_{22},$ and ${PSL}_{3} (F_{4}) \cdot 2_{2}$ over $Q$ », 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 $M_{12}$ über $Q (\sqrt{- 5})$ », Arch. Math. (Basel) 40 (1983), no. 3, p. 245-254. | DOI | Zbl

[16] B. H. Matzat & A. Zeh-Marschke, « Realisierung der Mathieugruppen $M_{11}$ und $M_{12}$ als Galoisgruppen über $Q$ », J. Number Theory 23 (1986), no. 2, p. 195-202. | DOI | Zbl

[17] J.-F. Mestre, « Construction d’extensions régulières de $Q (t)$ à groupes de Galois ${SL}_{2} (F_{7})$ et ${\tilde{M}}_{12}$ », 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 $M_{24}$ over $ℚ (t)$ », , 2012. | arXiv

[19] B. Plans & N. Vila, « Galois covers of $ℙ^{1}$ over $ℚ$ 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 $A B C$ 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: