On montre qu’il n’existe pas de corps de nombres primitif de degré ramifié en un unique premier petit. Il n’existe donc pas de corps de nombres de degré ramifié en un unique premier petit et ayant un groupe de Galois non résoluble.
We prove that there is no primitive nonic number field ramified only at one small prime. So there is no nonic number field ramified only at one small prime and with a nonsolvable Galois group.
@article{JTNB_2006__18_3_617_0,
author = {Lesseni, Sylla},
title = {Nonsolvable nonic number fields ramified only at one small prime},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {617--625},
publisher = {Universit\'e Bordeaux 1},
volume = {18},
number = {3},
year = {2006},
doi = {10.5802/jtnb.562},
mrnumber = {2330431},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.562/}
}
TY - JOUR AU - Lesseni, Sylla TI - Nonsolvable nonic number fields ramified only at one small prime JO - Journal de théorie des nombres de Bordeaux PY - 2006 SP - 617 EP - 625 VL - 18 IS - 3 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.562/ DO - 10.5802/jtnb.562 LA - en ID - JTNB_2006__18_3_617_0 ER -
%0 Journal Article %A Lesseni, Sylla %T Nonsolvable nonic number fields ramified only at one small prime %J Journal de théorie des nombres de Bordeaux %D 2006 %P 617-625 %V 18 %N 3 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.562/ %R 10.5802/jtnb.562 %G en %F JTNB_2006__18_3_617_0
Lesseni, Sylla. Nonsolvable nonic number fields ramified only at one small prime. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 617-625. doi : 10.5802/jtnb.562. http://www.numdam.org/articles/10.5802/jtnb.562/
[1] S. Brueggeman, Septic Number Fields Which are Ramified Only at One Small Prime. J. Symbolic Computation 31 (2001), 549–555. | MR | Zbl
[2] G. Butler, J. Mckay, The transitive groups of degree up to eleven. Comm. Algebra 11 (1983), no. 8, 863–911. | MR | Zbl
[3] F. Diaz Y Diaz, M. Olivier, Imprimitive ninth-degree fields with small discriminants. Math. Comp. 64 (1995), no. 209, 305–321. | MR | Zbl
[4] F. Diaz Y Diaz, Tables minorant la racine n-ième du discriminant d’un corps de nombres de degré n. Publications Mathématiques d’Orsay 80.06 (1980). | Zbl
[5] Y. Eichenlaub, Problèmes effectifs de théorie de Galois en degré 8 à 11. Thèse soutenue à l’université de Bordeaux 1 en 1996.
[6] ftp://megrez.math.u-bordeaux.fr/pub/pari/
[7] B. Gross, Modular forms (mod p) and Galois representation. Inter. Math. Res. Notices 16 (1998), 865–875. | MR | Zbl
[8] J. Jones, D. Roberts, Sextic number fields with discriminant . In Number Theory : Fifth Conference of the Canadian Number Theory Association, CRM Proceedings and Lecture Notes 19, 141–172. American Mathematical Society, 1999. | MR | Zbl
[9] S. Lesseni, The non-existence of nonsolvable octic number fields ramified only at one small prime. Mathematics of Computation 75 no. 255 (2006), 1519–1526. | MR | Zbl
[10] S. Lesseni, http://www.math.unicaen.fr/~lesseni/index.html. See “ma thèse”.
[11] J. Martinet, Petits discriminants des corps de nombres. In Number theory days, 1980 (Exeter, 1980), volume 56 of London Math. Soc. Lecture Note Series, 151–193. Cambridge Uni. Press, Cambridge, 1982. | MR | Zbl
[12] M. Pohst, On the computation of number fields of small discriminants including the mininum discriminants of sixth degree fields. J. Number Theory 14 (1982), 99–117. | MR | Zbl
[13] S. Selmane, Odlyzko-Poitou-Serre lower bounds for discriminants for number fields. Maghreb Math. Rev. 8 (1999), no 18.2. | MR
[14] J. P. Serre, Oeuvres, vol. . Springer-Verlag, Berlin/New York, 1986.
[15] K. Takeuchi, Totally real algebraic number fields of degree 9 with small discriminants. Saitama Math. J. 17 (1999), 63–85. | MR | Zbl
[16] J. Tate, The non-existence of certain Galois extension of unramified outside . Contemp. Math. 174 (1994) 153–156. | MR | Zbl
[17] R. Thompson, On the possible forms of discriminants of algebraic fields II. American J. of Mathematics 55 (1933), 110–118.
Cité par Sources :
