Constructing complete tables of quartic fields using Kummer theory

We explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using Kummer theory, instead of the traditional (and much less efficient) method using the geometry of numbers. As an application, we describe the computation of quartic fields of discriminant up to 107, the corresponding table being available by anonymous ftp.

[1]  Henri Cohen,et al.  Advanced topics in computational number theory , 2000 .

[2]  M. Olivier,et al.  The computation of sextic fields with a cubic subfield and no quadratic subfield , 1992 .

[3]  Henri Cohen,et al.  Tables of octic fields with a quartic subfield , 1999, Mathematics of Computation.

[4]  H. Davenport,et al.  On the Density of Discriminants of Cubic Fields , 1969 .

[5]  Karim Belabas,et al.  A fast algorithm to compute cubic fields , 1997, Math. Comput..

[6]  H. Davenport,et al.  On the density of discriminants of cubic fields. II , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Jacques Martinet,et al.  The computation of sextic fields with a quadratic subfield , 1990 .

[8]  Johannes Buchmann,et al.  On the computation of totally real quartic fields of small discriminant , 1989 .

[9]  Henri Cohen,et al.  Computing ray class groups, conductors and discriminants , 1996, Math. Comput..

[10]  Johannes Buchmann,et al.  Enumeration of quartic fields of small discriminant , 1993 .

[11]  D. Bressoud,et al.  A Course in Computational Number Theory , 2000 .

[12]  Henri Cohen,et al.  Construction of Tables of Quartic Number Fields , 2000, ANTS.

[13]  D. Ford.  Enumeration of totally complex quartic fields of small discriminant , 1991 .

[14]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[15]  Manjul Bhargava,et al.  Gauss Composition and Generalizations , 2002, ANTS.