Advanced Topics in Computational Number Theory: 193 (Graduate Texts in Mathematics, 193) - Hardcover

Cohen, Henri

Hardcover

ISBN 10: 0387987274 ISBN 13: 9780387987279

Publisher: Springer, 1999

Synopsis

The computation of invariants of algebraic number fields such as integral bases, discriminants, prime decompositions, ideal class groups, and unit groups is important both for its own sake and for its numerous applications, for example, to the solution of Diophantine equations. The practical com­ pletion of this task (sometimes known as the Dedekind program) has been one of the major achievements of computational number theory in the past ten years, thanks to the efforts of many people. Even though some practical problems still exist, one can consider the subject as solved in a satisfactory manner, and it is now routine to ask a specialized Computer Algebra Sys­ tem such as Kant/Kash, liDIA, Magma, or Pari/GP, to perform number field computations that would have been unfeasible only ten years ago. The (very numerous) algorithms used are essentially all described in A Course in Com­ putational Algebraic Number Theory, GTM 138, first published in 1993 (third corrected printing 1996), which is referred to here as [CohO]. That text also treats other subjects such as elliptic curves, factoring, and primality testing. Itis important and natural to generalize these algorithms. Several gener­ alizations can be considered, but the most important are certainly the gen­ eralizations to global function fields (finite extensions of the field of rational functions in one variable overa finite field) and to relative extensions ofnum­ ber fields. As in [CohO], in the present book we will consider number fields only and not deal at all with function fields.

About this title

Review

"Das vorliegende Buch ist eine Fortsetzung des bekannten erkes "A Course in Computational Algebraic Number Theory" (Graduate Texts in Mathematics 138) desselben Autors. ...
So ist das vorliegende Buch ein sehr umfängliches Nachschlagewerk zur algorithmischen Zahlentheorie, das zusammen mit dem ersten Buch des Autors sicherlich eine Standard-Referenz für zahlentheoretische Algorithmen darstellen wird."
Internationale Mathematische Nachrichten, Nr. 187, August 2001

Synopsis

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; and 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.

Edition details

• Publisher: Springer
• Publication date: 1999
• Language: English
• ISBN 10: 0387987274
• ISBN 13: 9780387987279
• Binding: Hardcover
• Number of pages: 596