Computational algebraic number theory has been attracting broad interest in the last few years due to its potential applications in coding theory and cryptography. For this reason, the Deutsche Mathematiker Vereinigung initiated an introductory graduate seminar on this topic in Dรผsseldorf. The lectures given there by the author served as the basis for this book which allows fast access to the state of the art in this area. Special emphasis has been placed on practical algorithms - all developed in the last five years - for the computation of integral bases, the unit group and the class group of arbitrary algebraic number fields. Contents: Introduction โข Topics from finite fields โข Arithmetic and polynomials โข Factorization of polynomials โข Topics from the geometry of numbers โข Hermite normal form โข Lattices โข Reduction โข Enumeration of lattice points โข Algebraic number fields โข Introduction โข Basic Arithmetic โข Computation of an integral basis โข Integral closure โข Round-Two-Method โข Round-Four-Method โข Computation of the unit group โข Dirichlet's unit theorem and a regulator bound โข Two methods for computing r independent units โข Fundamental unit computation โข Computation of the class group โข Ideals and class number โข A method for computing the class group โข Appendix โข The number field sieve โข KANT โข References โข Index
CONTENT
Intorduction -- Topics from finite field -- Topics from the geometry of number -- Algebraic number field -- Computation of an integral basis -- Computation of the unit group -- Computation of the class group -- ยง 1 The number field sieve -- ยง 2 KANT -- References
