Ebook: Advanced Topics in Computational Number Theory

Fundamental Results and Algorithms in Dedekind domains -- Basic Relative Number Field Algorithms -- The Fundamental Theorems of Global Class Field Theory -- Computational Class Field Theory -- Computing Defining Equations -- Cubic Number Fields -- Ramification, Conductors and Discriminants -- Relative Class Groups, Units and Regulators -- Inverting Prime Ideals -- Algorithms for p-adic fields.;The present book addresses a number of specific topics in computational number theory whereby the author is not attempting to be exhaustive in the choice of subjects. The book is organized as follows. Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms. Chapters 3, 4, and 5 contain the theory and complete algorithms concerning class field theory over number fields. The highlights are the algorithms for computing the structure of (Z_K/m)^*, of ray class groups, and relative equations for Abelian extensions of number fields using Kummer theory. Chapters 1 to 5 form a homogeneous subject matter which can be used for a 6 months to 1 year graduate course in computational number theory. The subsequent chapters deal with more miscellaneous subjects. Written by an authority with great practical and teaching experience in the field, this book together with the author's earlier book will become the standard and indispensable reference on the subject.