Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (Z K /m) * for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

[1]  Henri Cohen,et al.  A polynomial reduction algorithm , 1991 .

[2]  Armin Leutbecher,et al.  Euclidean fields having a large Lenstra constant , 1985 .

[3]  Andrew Odlyzko,et al.  Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results , 1990 .

[4]  Gerhard Niklasch,et al.  On cliques of exceptional units and Lenstra's construction of Euclidean fields , 1989 .

[5]  Jacques Martinet,et al.  Journées Arithmétiques 1980: Petits discriminants des corps de nombres , 1982 .

[6]  Xavier-François Roblot,et al.  Unités de Stark et corps de classes de Hilbert , 1996 .

[7]  Henri Cohen,et al.  Hermite and Smith normal form algorithms over Dedekind domains , 1996, Math. Comput..

[8]  George Havas,et al.  Hermite normal form computation for integer matrices , 1994 .

[9]  Michael E. Pohst,et al.  Computations with relative extensions of number fields with an application to the construction of Hilbert class fields , 1995, ISSAC '95.

[10]  Michael Pohst,et al.  Algorithmic algebraic number theory , 1989, Encyclopedia of mathematics and its applications.

[11]  Hans Zantema,et al.  Class numbers and units , 1980 .

[12]  Norikata Nakagoshi.  The structure of the multiplicative group of residue classes modulo ${\germ p}^{N+1}$ , 1979 .

[13]  J. Neukirch.  Algebraic Number Theory , 1999 .

[14]  E. Hecke.  Lectures on the Theory of Algebraic Numbers , 1981 .

[15]  O. H. Lowry.  Academic press. , 1972, Analytical chemistry.

[16]  Jeffrey Shallit,et al.  Factor refinement , 1993, SODA '90.

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