Septic fields with discriminant ±2^{𝑎}3^{𝑏}

Jones, John; Roberts, David

doi:10.1090/S0025-5718-03-01510-2

Septic fields with discriminant $\pm 2^a 3^b$
HTML articles powered by AMS MathViewer

by John W. Jones and David P. Roberts PDF

Math. Comp. 72 (2003), 1975-1985 Request permission

Abstract:

We classify septic number fields which are unramified outside of $\{\infty ,2, 3\}$ by a targeted Hunter search; there are exactly $10$ such fields, all with Galois group $S_7$. We also describe separate computations which strongly suggest that none of these fields come from specializing septic genus zero three-point covers.

References

Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Sharon Brueggeman, Septic number fields which are ramified only at one small prime, J. Symbolic Comput. 31 (2001), no. 5, 549–555. MR 1828702, DOI 10.1006/jsco.2001.0440
Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
David Harbater, Galois groups with prescribed ramification, Arithmetic geometry (Tempe, AZ, 1993) Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 35–60. MR 1299733, DOI 10.1090/conm/174/01850
Hermite, C., Sur le nombre limité d’irrationalités auxquelle se réduisent les racines des équations à coefficients entiers complexes d’un degré et d’un discriminant donnés (Extrait d’une lettre à M. Borchardt), J. Reine Angew. Math. 53, (1857), 182–192 = Oeuvres, I, Paris 1905 414–428.
Jones, J., Tables of number fields with prescribed ramification, http://math.la.asu.edu/~jj/numberfields
John W. Jones and David P. Roberts, Sextic number fields with discriminant $(-1)^j2^a3^b$, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 141–172. MR 1684600, DOI 10.1090/crmp/019/16
John W. Jones and David P. Roberts, Timing analysis of targeted Hunter searches, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 412–423. MR 1726089, DOI 10.1007/BFb0054880
Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
Gunter Malle, Genus zero translates of three point ramified Galois extensions, Manuscripta Math. 71 (1991), no. 1, 97–111. MR 1094741, DOI 10.1007/BF02568396
Gunter Malle, Fields of definition of some three point ramified field extensions, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 147–168. MR 1305396
Michael Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), no. 1, 99–117. MR 644904, DOI 10.1016/0022-314X(82)90061-0
Roberts, D., An ABC construction of number fields, preprint, http://cda.mrs.umn.edu/~roberts