A targeted Martinet search

Driver, Eric; Jones, John

doi:10.1090/S0025-5718-08-02178-9

A targeted Martinet search
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by Eric D. Driver and John W. Jones;

Math. Comp. 78 (2009), 1109-1117

DOI: https://doi.org/10.1090/S0025-5718-08-02178-9

Published electronically: August 22, 2008

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Abstract:

Constructing number fields with prescribed ramification is an important problem in computational number theory. In this paper, we consider the problem of computing all imprimitive number fields of a given degree which are unramified outside of a given finite set of primes $S$ by combining the techniques of targeted Hunter searches with Martinet’s relative version of Hunter’s theorem. We then carry out this algorithm to generate complete tables of imprimitive number fields for degrees $4$ through $10$ and certain sets $S$ of small primes.

References

BOINC, Berkeley Open Infrastructure for Network Computing. http://boinc.berkeley.edu
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
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
F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré $8$, J. Number Theory 25 (1987), no. 1, 34–52 (French, with English summary). MR 871167, DOI 10.1016/0022-314X(87)90014-X
F. Diaz y Diaz and M. Olivier, Imprimitive ninth-degree number fields with small discriminants, Math. Comp. 64 (1995), no. 209, 305–321. With microfiche supplement. MR 1260128, DOI 10.1090/S0025-5718-1995-1260128-X
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, Septic fields with discriminant $\pm 2^a3^b$, Math. Comp. 72 (2003), no. 244, 1975–1985. MR 1986816, DOI 10.1090/S0025-5718-03-01510-2
J. Jones, Tables of number fields with prescribed ramification, http://math.la.asu.edu/~jj/numberfields
E. Driver, Tables of number fields with prescribed ramification, http://hobbes.la.asu.edu/Number_Fields/Driver/FieldTables.html
E. Driver, A Targeted Martinet Search, Ph.D. Thesis, Arizona State University, December 2006.
Lesseni Sylla, The nonexistence of nonsolvable octic number fields ramified only at one small prime, Math. Comp. 75 (2006), no. 255, 1519–1526. MR 2219042, DOI 10.1090/S0025-5718-06-01827-8
Sylla Lesseni, Nonsolvable nonic number fields ramified only at one small prime, J. Théor. Nombres Bordeaux 18 (2006), no. 3, 617–625 (English, with English and French summaries). MR 2330431
Jacques Martinet, Methodes géométriques dans la recherche des petits discriminants, Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, pp. 147–179 (French). MR 902831
PARI2, 2000. PARI/GP, Version 2.1.4. The PARI Group, Bordeaux. http://www.parigp-home.de
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
M. Pohst, J. Martinet, and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), no. 2, 145–159. MR 1072461, DOI 10.1016/0022-314X(90)90069-4
Schehrazad Selmane, Non-primitive number fields of degree eight and of signature $(2,3)$, $(4,2)$ and $(6,1)$ with small discriminant, Math. Comp. 68 (1999), no. 225, 333–344. MR 1489974, DOI 10.1090/S0025-5718-99-00998-9
Schehrazad Selmane, Quadratic extensions of totally real quintic fields, Math. Comp. 70 (2001), no. 234, 837–843. MR 1697649, DOI 10.1090/S0025-5718-00-01210-2
Schehrazad Selmane, Tenth degree number fields with quintic fields having one real place, Math. Comp. 70 (2001), no. 234, 845–851. MR 1709158, DOI 10.1090/S0025-5718-00-01232-1