Nonsolvable nonic number fields ramified only at one small prime
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 3, pp. 617-625.

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.

On montre qu’il n’existe pas de corps de nombres primitif de degré 9 ramifié en un unique premier petit. Il n’existe donc pas de corps de nombres de degré 9 ramifié en un unique premier petit et ayant un groupe de Galois non résoluble.

DOI: 10.5802/jtnb.562
Keywords: Nonic field. Galois group. Nonsolvable
Sylla Lesseni 1

1 Université Bordeaux 1 Laboratoire d’Algorithmique Arithmétique 351, Cours de la Libération 33405 Talence Cedex, France.
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Sylla Lesseni. Nonsolvable nonic number fields ramified only at one small prime. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 3, pp. 617-625. doi : 10.5802/jtnb.562. https://jtnb.centre-mersenne.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 (-1) j 2 a 3 b . 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. 3. 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 2. 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.

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